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LIBRARY 

OF  THK 

UNIVERSITY  OF  CALIFORNIA. 

01  FT  OK 

PACIFIC  THEOLOGICAL  SEMINARY. 


^Accession 


Class 


UNIVERSITY   EDITION. -REVISED   AND   ENLARGED. 


TREATISE 


ON 


SURVEYING  AND  NAVIGATION 


DNITINO 


THE  THEORETICAL,  PRACTICAL,  AND 


EDUCATIONAL  FEATURES  OF  THESE  SUBJECTS, 


BY  HORATIO  N.  ROBINSON,  A.  M., 
ki 

FORMERLY   PROFESSOR   OF   MATHEMATICS   IN   THK    UNITED   STATES    NAVY;   AUTHOR  OF 
MATHEMATICAL,   PHILOSOPHICAL,    AND   ASTRONOMICAL   WORKS. 


FOURTH    STANDARD    EDITION. 


NEW    YORK: 

IVISON    &    PHINNEY,    321    BROADWAY 

CINCINNATI:      JACOB      ERNST. 
CHICAGO :   8.  C.  GRIGGS  &  CO.,  89  <fc  41  LAKE  ST. 

ST.  LOCIS:   KEITH  k  WOODS.    BUFFALO  I   PHINNEY  fc  CO. 


1868. 


ENTERED,  according  to  act  of  Congress,  in  the  year  1852, 

BY  H.  N.  ROBINSON, 
in  the  Clerk's  Office  of  the  District  Court  for  the  Northern  District  of  New  York. 


Entered  according  to  act  of  Congress,  in  the  year  ICStf, 
BY  H.  N.  ROBINSON, 

in  the  Clerk'i  Office  of  the  District  <3o«f  fa  the  Northern  District  of  New  York. 


A.   C.   JAMES,  STEREOTYPEB, 

167   WALNUT   ST.,    CINCINNATI 


PREFACE. 


THIS  book  is  more  than  its  title  page  proclaims  it  to  be :  it  is  the 
practical  application  of  the  Mathematical  Sciences  to  Mensuration,  to 
Land  Surveying,  to  Leveling,  and  to  Navigation. 

Nor  is  the  work  merely  practical.  Elementary  principles  are  here 
and  there  brought  before  the  mind  in  a  new  light ;  and  original  investi- 
gations will  be  found  in  many  parts  of  the  work.  To  show  the  reader 
how  a  thing  is  to  be  done,  is  but  a  small  part  of  the  object  sought  to  be 
obtained :  the  great  stress  is  put  upon  the  reasons  for  so  doing,  which 
gives  true  discipline  to  the  mind,  and  adds  greatly  to  the  educational 
value  of  any  book. 

We  have  illustrated  the  subject  of  logarithms,  and  their  practical 
uses,  the  same  in  this  book  as  is  common  to  be  found  in  other  books,  and 
this  is  sufficient  for  the  common  pupil,  or  the  ordinary  practical  man, 
whether  surveyor  or  navigator  ;  but  in  addition  to  this,  we  have  carried 
logarithms  much  further  in  this  work  than  in  any  I  have  seen.  I  do 
not  mean  by  this  that  we  have  more  voluminous  tables  than  others. 
Such  is  not  the  fact. 

Voluminous  tables  are  not  necessary  for  those  who  really  understand 
the  nature  of  logarithms,  and  such  are  mainly  intended  for  those  who 
are  not  expected  to  understand  principles.  To  give  a  more  practical 
illustration  of  logarithms,  and  to  suggest  artifices  in  using  logarithms 
generally,  we  have  given  Table  III  and  its  auxiliaries,  on  page  70  of 
tables,  showing  logarithms  to  twelve  places  of  decimals,  a  degree  of 
accuracy  which  practice  never  demands.  By  the  help  of  this  table 
combined  with  a  true  knowledge  of  the  subject,  the  logarithm  of  any 
number  may  be  readily  found  true  to  ten  places  of  decimals,  or,  conversely 
the  number  corresponding  to  any  given  logarithm  may  be  found  to 
almost  any  degree  of  accuracy. 

Our  Traverse  Table  is  not  so  full  as  in  some  other  books,  but  it 

1 11898 


T  PREFACE. 

is  full  enough  to  answer  every  purpose ;  and  latitude  and  departure, 
corresponding  to  any  course  and  distance,  can  be  found  by  it,  provided 
the  operator's  good  judgment  is  awake.  Indeed  a  contracted  table,  in 
an  educational  point  of  view,  is  better  than  a  full  one  ;  for  the  former 
calls  forth  and  cultivates  tact  in  the  student,  but  the  latter  is  best  for 
the  unanimated  plodder. 

In  running  lines,  and  computing  the  areas  of  surveys,  we  have 
endeavored  to  present  the  subject  in  such  a  manner  that  the  reader  must 
constantly  keep  Elementary  Geometry  in  view,  and  the  whole  is  so 
clear  and  simple,  that  many  will  think  it  unworthy  of  the  rank  that  it 
seems  to  hold  in  the  public  estimation,  but  there  are  other  reasons  for 
this. 

The  chapter  on  surveys  and  surveyors  will  be  found  to  be  a  little 
peculiar,  but  the  information  there  given,  will  be  highly  useful  to  all 
those  who  are  inclined  to  look  upon  a  survey  as  a  mathematical  problem 
only. 

On  the  compass,  and  the  declination  of  the  needle,  we  have  been 
very  full :  the  subject  embraces  meridians  and  astronomical  lines  drawn 
on  the  earth. 

The  manner  in  which  we  should  proceed  to  make  a  survey,  provided 
no  such  instrument  as  the  compass  existed,  and  there  were  no  such 
thing  as  a  magnetic  needle,  is  taken  up  and  illustrated  in  this  work. 

The  subject  of  dividing  lands  is  fully  discussed  and  illustrated,  and 
if  any  one  has  occasion  to  complain  of  mathematical  abstrusity  in  this 
work,  it  will  be  found  in  this  connection ;  yet  there  is  nothing  here 
above  elementary  algebra  and  geometry. 

The  method  of  taking  levels  and  making  a  profile  of  the  vertical 
section  of  a  line  for  rail  roads,  is  set  forth  in  this  work.  The  profile 
shows  the  necessary  excavation  or  embankment,  which  it  is  necessary 
to  cut  down  or  build  up  at  any  particular  point,  to  conform  to  any 
proposed  grade  that  may  be  contemplated. 

To  determine  the  elevation  of  any  place  above  the  level  of  the  sea, 
by  means  of  the  barometer,  has  been,  and  now  is,  a  very  interesting  pro- 
blem to  all  philosophical  students,  yet  very  few  of  them  have  been  able 
to  comprehend  it  beyond  its  first  great  principle,  the  variation  of  atmos- 
pheric pressure.  To  trace,  or  rather  to  discover  the  mathematical  law 
which  connects  the  elevation  of  any  locality  with  the  mean  hight  of  the 
barometer  at  the  same  place,  has  been  an  obscure  problem,  and  we  have 
taken  hold  of  it  with  a  determination  to  break  open  some  avenue  of 
light  (if  such  were  possible)  by  which  the  simplicity  of  the  problem 
might  be  brought  to  the  comprehension  of  the  every-day  mathematical 
student,  and  we  believe  that  we  have  succeeded  in  the  undertaking. 


PREFACE. 


The  part  on  Navigation,  might  be  regarded,  at  first  view,  an  abridg- 
ment of  that  subject,  and  in  one  sense  it  is,  for  we  have  studied  to  be 
as  brief  as  possible,  but  we  would  never  let  brevity  stand  in  the  place  of 
perspicuity;  and  however  it  may  appear,  we  have  given  all  the 
mathematical  essentials  of  the  subject,  and  whoever  acquires  what  is 
here  given,  will  find  very  little  necessity  of  looking  elsewhere  for  the 
continuation  of  the  study,  unless  it  is  for  sea  terms  and  seamanship  ; 
but  these  have  nothing  to  do  with  Navigation  as  a  science.  Our  method 
of  working  lunars  is  more  brief  than  any  other,  where  auxiliary  tables 
and  methods  of  approximation  are  not  resorted  to,  but  to  attain  this 
brevity,  we  have  been  compelled  to  use  Natural  Sines  in  part  of  the 
operation  ;  but  on  the  other  hand,  this  should  be  no  objection,  for  it 
gives  us  a  clearer  view  of  the  unity  and  harmony  of  the  mathematical 


sciences. 


PREFACE   TO   THE  FOURTH  EDITION. 


For  reasons  which  we  have  not  here  the  space  to  explain,  we  have  thought 
best  to  remove  the  matter  between  the  32d  and  43d  pages  of  the  former  edi- 
tions, to  the  last  chapter  in  the  book :  to  fill  up  that  space  with  more  simple 
and  more  practical  matter,  and  to  enrich  the  volume  with  additional  pages 
containing  very  choice  miscellaneous  matter.  We  are  induced  to  make  these 
improvements,  by  the  strong  conviction  that  this  work  contains  all  the  essen- 
tials of  popularity  and  permanency. 

Objections  have  been  made  to  the  brevity  of  our  traverse  table,  and  at  one 
time  we  thought  of  enlarging  it  —  but  on  further  reflection,  we  concluded 
that  this  was  a  mere  objection,  given  out  for  the  want  of  a  better  one.  This 
is  designed  as  an  educational  volume,  and  a  properly  educated  person  does 
not  require  a  voluminous  traverse  table.  Such  tables  are  mostly  intended 
for  those  who  do  not  pretend  to  understand  them,  and  they  are  really  required 
only  for  about  one  in  a  thousand  of  those  who  study  this  subject.  The  sur- 
veyor who  is  in  constant  practice,  and  such  persons,  have  tables  separate 
from  all  other  matter,  in  such  a  form  as  to  roll  up  and  carry  in  the  pocket. 


CONTENTS. 

INTRODUCTION. 

CHAPTER  I. 

PAOB 

Introduction 9 

Construction  of  Geometrical  Problems,  with  the  use  of  instru- 
ments    12 

CHAPTER  II. 

Logarithms 22 

Application  of  Logarithms  to  Multiplication,  Division,  and  the 

extraction  of  Roots 29 — 32 

Artifices  in  the  use  of  Logarithms  (Art.  12) 32—34 

Logarithms  without  the  use  of  a  table 34—43 

CHAPTER  III. 

Plane  Trigonometry 44 

Explanation  of  Tables 58 

Oblique-angled  Plane  Trigonometry 64 

SURVEYING. 

Introductory  Remarks 70 

Finding  Areas  in  general .  70 — 79 

Mensuration  of  Solids 79 

CHAPTER  I. 

Mensuration  of  Lands 80 

To  measure  a  Line 81 

Surveyor's  Compass 82 — 85 

Vernier  Scales  in  general 85 — 86 

CHAPTER   II. 

Latitude  and  Departure 87 

Taking  angles  by  the  Compass 89 

(vi) 


CONTENTS.  vii 

PAGE. 

To  close  a  Survey  ...................  89 

To  find  the  true  from  a  random  Line  ...........  90 

Computation  of  Areas  by  trapezoids  ...........  91  —  103 

CHAPTER  III. 

To  find  Meridian  Lines  .................  103 

Variation  of  the  Compass  ...............  104  —  1  12 

Practical  Difficulties  ..................  112 

CHAPTER  IV. 

To  Survey  without  a  Compass  .............  113 

The  Circumferentor  ..................  114 

CHAPTER  V. 

Original  and  subsequent  Surveys  ............  116 

Difficulties  and  Duties  of  a  Surveyor  ...........  118 

United  States'  Land.   .  '  ................  121 

CHAPTER  VI. 

Very  irregular  Figures  .................  123 

Division  of  Lands  —  a  variety  of  Problems   ........  124  —  145 

CHAPTER  VII. 

Triangular  Surveying  .................  146 

The  Plane  Table—  its  uses  &c  ..............  147—152 

152 


Piloting  Ships    ...................  154 

CHAPTER  VIII. 

Leveling  .......................  155 

Description  of  the  Level  ................  158 

Adjusting  the  Level  ..................  159  —  161 

Keeping  Book    ....................  163 

Contour  of  Ground  ..................  165 

Elevation  determined  by  atmospheric  Pressure,  as  indicated  by 

the  Barometer   ...................  166  —  174 

NAVIGATION. 

CHAPTER  I. 

Introduction  .....................  175 

The  Log-line  .................   ....  176 

The  Mariner's  Compass  ................  177 


viii  CONTENTS. 

CHAPTER  II. 

PAG*. 

Plane  Sailing 180—186 

Middle  Latitude  Sailing 186 

Traverse  Sailing 186 

Sailing  in  Currents 191 

CHAPTER   III. 

Mercator's  Chart — its  construction 194 — 196 

Mercator's  Sailing 196 

CELESTIAL    OBSERVATIONS. 
CHAPTER  I. 

Definition  of  Terms 198—199 

Quadrant  and  Sextant 202 

Construction  of  the  Sextant 204 

The  adjustments  of  the  Sextant 205 

To  take  an  Altitude  of  the  Sun  at  sea 206 

To  find  the  Latitude  by  the  Sun  or  Stars 207 

To  find  the  Latitude  by  the  meridian  Altitude  of  the  Moon  .  209—211 

CHAPTER   II. 

A  perfect  Time-piece .' 213 

Local  Time— Rule  to  find  it 216 

Longitude  by  Chronometer 217 

CHAPTER   III. 
Lunar  Observations 220 

Formulae,  for  clearing  the  observed  Distance  from  the  effects  of 

Parallax  and  Refraction 223 

Examples  for  working  Lunars 224 — 227 

APPENDIX. 

Artifices  to  be  resorted  to  in  difficult  circumstances    ....  228 

Results  corresponding  to  assumed  errors  of  observation 231 

Application  of  the   differential  calculus,    to  find  results  corres- 
ponding to  eirors  of  observation 232 

Application  of  the  same,  to  clearing  lunar  distances 233 

Change  of  the  formulas,  to  avoid  the  use  of  Natural  sines  and  Nat- 
ural cosines 235 

Artifices,  in  the  use  and  computation  of  logarithms 239 — 246 


INTBODUCTION 


CHAPTER    I. 

MENSURATION,  SURVEYING,  and  NAVIGATION,  are  but  branches  of  the 
same  science,  and  should  be  regarded  as  the  application  of  geometry 
and  trigonometry,  and  in  this  light  we  shall  present  them  to  our 
readers. 

In  this  volume  we  shall  not  demonstrate  geometrical  truths  unless 
we  wish  to  present  them  in  some  new  form,  or  unless  the  demon- 
stration is  not  readily  to  be  found  in  the  proper  places,  in  the  elemen- 
tary books. 

It  is  expected  that  all  readers  of  a  work  of  this  kind,  have 
previously  made  themselves  more  or  less  acquainted  with  Algebra 
and  Geometry,  and  where  this  is  the  case  the  reader  will  have  no 
difficulty  ;  and  readers  who  are  not  thus  prepared  should  be  careful 
not  to  charge  imaginary  defects  to  the  book  :  in  no  work  of  this 
kind  would  it  be  proper  to  demonstrate  every  elementary  principle. 
These  remarks  apply  only  to  the  educational  character  of  the  book. 

Preparatory  to  a  course  of  practical  mathematics,  it  is  proper  to 
give  such  descriptions  of  the  instruments  to  be  used  as  will  enable 
the  operator  to  understand  their  use.  But  some  of  these  instru- 
ments can  never  be  understood  from  a  book,  it  must  be  from  the 
instrument  itself ;  we  might  as  well  attempt  to  give  a  person  an  idea 
of  color  by  the  means  of  language,  as  to  give  a  person  a  correct  idea 
of  the  sextant  and  theodolite  by  a  mere  book  description.  It  is  true 
we  can  do  something  by  drawings  and  descriptions,  and  that  some- 
thing we  intend  to  do. 

To  represent  plane  surfaces  and  tracts  of  land  on  paper,  no  other 
instruments  are  necessary  than  the  scale,  and  dividers,  and  a  pro- 

(9) 


10  SURVEYING. 

tractor  to  measure  angles.  In  fact,  every  thing  can  be  done  with 
the  scale  and  dividers  only  —  other  instruments,  as  the  protractor, 
sector,  and  parallel  rulers,only  add  to  our  convenience ;  at  the  same 
time  they  could  be  dispensed  with. 


THE      PLANE     SCALE. 


The  plane  scale,  or  the  plane  diagonal  scale  of  equal  parts,  as  here 
represented,  is  the  most  common  and  useful  of  all  the  instruments 
used  in  drawing.  It  is  also  a  ruler,  and  if  wide  and  well  made  will 
serve  as  a  square  also,  by  which  right  angles  may  be  drawn. 


The  very  appearance  of  this  scale  will  show  its  construction,  the 
side  of  the  square  a  b  may  be  of  any  length  whatever,  it  is  gener- 
ally taken  an  inch,  but  this  is  not  imperative. 

By  means  of  the  10  parallel  lines  running  along  the  length  of  the 
scale,  and  the  10  diagonal  lines  parallel  to  each  other  in  the  square 
a  b  c  d,  we  have  100  intersections  in  the  square,  by  which  we  are 
enabled  to  find  any  and  every  hundredth  part  of  the  division  of  a  b. 

For  example,  I  wish  to  find  27  hundredths  of  the  line  a  b.  I  go 
to  the  division  2  on  a  b,  and  then  run  up  that  diagonal  line  to  the  7th 
parallel,  and  from  that  intersection  to  the  line  a  d  is  27  hundredths 
of  a  b. 

The  distances  a  b,  a  g,  g  h,  &c.,  may  be  taken  to  represent  1,10, 
100,  or  in  fact  any  number  we  please.  Suppose  we  take  any  one 
of  the  equal  divisions  a  b,  ag,  <fec.,  to  represent  100,  and  then  require 
234.  From  I  to  e  represents  that  distance. 

If  the  base  a  b  were  10,  from  I  to  e  would  be  23.4  ;  if  1,  then  from 
I  to  e  would  be  2.34 ;  and  so  on  proportionally  for  any  other  change 
of  base,  or  change  of  the  unit. 

To  transfer  distances  from  the  scale  (as  I  e,  p  g,  <fec.)  to  paper, 
we  require 


INTRODUCTION.  11 

DIVIDERS. 

Dividers  are  nothing  more  than  a  delicate  pair  of  compasses  — 
two  bars  turning  on  a  joint.  They  are  too  well  known  to  require 
representation  by  a  figure. 

They  are  also  used  for  describing  circles  and  parts  of  circles. 

THE     PROTRACTOB. 

The  following  diagram  accurately  represents  this  instrument.  It 
consists  of  a  semicircle  of  brass  ABC,  divided  into  degrees. 


The  degrees  are  numbered  both  ways,  from  A  to  B  and  from  B 
to  A.  There  is  a  small  notch  in  the  middle  of  AB,  to  indicate  the 
center. 

To  lay  off  an  angle.  Place  the  diameter  AB  on  the  line,  so  that 
the  center  shall  fall  on  the  angular  point. 

Then  at  the  degree  required,  at  the  edge  of  the  semicircle  make 
a  point  with  a  pin.  Then  remove  the  protractor  and  draw  a  line 
through  the  point  so  marked  and  the  angular  point ;  this  line,  with 
the  given  line,  will  make  the  required  angle. 

The  reader  will  observe  a  great  similarity  between  this  instrument 
and  the  circumferentor,  which  is  described  in  a  subsequent  portion 
of  this  work. 

This  instrument  is  designed  merely  to  draw  angles  on  paper,  that 
to  draw  lines  marking  given  angles,  with  other  lines,  in  the  field. 


12  SURVEYING. 

In  addition  to  this,  both  the  protractor  and  circumferentor  may  be 
used  in  taking  levels,  and  measuring  angles  of  altitude,  when  no 
better  instruments  for  such  purposes  are  at  hand. 

For  instance,  if  a  delicate  plumb  should  be  suspended  from  the 
center  of  the  protractor,  and  the  thread  rest  at  the  point  (7,  while  the 
instrument  is  held  in  a  frame,  then  A  and  B  would  be  as  a  level, 
and  as  many  degrees  as  the  plumb  line  rested  from  C  so 
many  degrees  would  be  the  inclination  of  A  and  B  from  a  horizontal 
level. 

Levels  and  angles  of  altitudes  were  formerly  taken  in  this  way. 

"With  the  instruments  previously  described,  solve  the  following 
problems.  The  references  are  to  Robinson's  Geometry.  Thus, 
(th.  15,  b.  1,  cor.  1,)  indicates  theorem  15,  book  1,  corollary  1, 
where  the  demonstrations  of  the  problem  referred  to  will  be  found. 

PROBLEM    1  . 

To  bisect  a  given  finite  straight  line. 

Let  AB  be  the  given  line,  and  from  its 
extremities,  A  and  B,  with  any  radius 
greater  than  the  half  of  AB  (Post  3),  de- 
scribe arcs,  cutting  each  other  in  n  and  m. 
Join  n  and  m;  and  (7,  where  it  cuts  ABf 
will  be  the  middle  of  the  line  required. 

Proof,  (th.  15,  b,  1,  cor.  1  ). 


PROBLEM    2. 

To  bisect  a  given  angle. 

Let  ABC  be  the  given  angle.  With  any 
radius,  from  the  center  B,  describe  the  arc 
AC.  From  A  and  C,  as  centers,  with  a 
radius  greater  than  the  half  of  A  (7,  de- 
scribe arcs,  intersecting  in  n;  and  join  Bn, 
it  will  bisect  the  given  angle. 

Proof,  (th.  19,  b.  1). 


INTRODUCTION. 


13 


PROBLEM    3. 

From  a  given  point,  in  a  given  line,  to  draw  a  perpendicular  to  that 
line. 

Let  AS  be  the  given  line,  and  0 
the  given  point.  Take  n  and  m  equal 
distances  on  opposite  sides  of  C;  and 
from  the  points  m  and  n,  as  centers, 
with  any  radius  greater  than  nC  or 
or  mC,  describe  arcs  cutting  each  other 
in  S.  Join  SO,  and  it  will  be  the  per- 
pendicular required.  Proof,  (th.  15,  b.  1,  cor.  ). 

The  following  is  another  method,  which 
is  preferable,  when  the  given  point,  C,  is  at 
or  near  the  end  of  the  line. 

Take  any  point,  0,  which  is  manifestly 
one  side  of  the  perpendicular,  and  join  0  C; 
and  with  OC,  as  a  radius,  describe  an  arc, 
cutting  AB  in  m  and  C.  Join  m  0,  and  produce  it  to  meet  the 
arc,  again,  in  n;  mn  is  then  a  diameter  to  the  circle.  Join  Cn,  and 
it  will  be  the  perpendicular  required.  Proof,  (th.  9,  b.  3). 


PROBLEM   4. 

From  a  given  point  unthout  a  line,  to  draw  a  perpendicular  to  that 
line. 

Let  AB  be  the  given  line,  and  C  the 
given  point.  From  C,  draw  any  oblique 
line,  as  Cn.  Find  the  middle  point  of 
Cn  by  (problem  1),  and  from  that  point, 
as  a  center,  describe  a  semicircle,  having 
Cn  as  a  diameter.  From  the  point  m, 
where  this  semicircle  cuts  AB,  draw  Cm, 
and  it  will  be  the  perpendicular  required.  Proof,  (th.  9,  b.  3). 


14  SURVEYING. 

PROBLEM    5. 

At  a  given  point  in  a  line,  to  make  an  angle  equal  to  another  given 
angle. 

Let  A  be  the  given  point  in  the  line  AB, 
and  DCE  the  given  angle. 

From   C  as  a  center,  with  any  radius, 
CE,  draw  the  arc  ED. 

From  A,  as  a    center,  with  the  radius 
AF—  CE,  describe  an  indefinite  arc ;  and 
from  F,  as  a  center,  with  FG  as  a  radius, 
equal  to  ED,  describe  an  arc,  cutting  the  other  arc  in  G,  and  join 
AG;  GAF  will  be  the  angle  required.    Proof,  (th.  5,  b.  3). 

PROBLEM    6. 

From  a  given  point,  to  draw  a  line  parallel  to  a  given  line. 

Let  A  be  the  given  point,  and  CB  the 
given  line.  Draw  AB,  making  an  angle, 
ABC;  and  from  the  given  point,  A,  in  the 
line  AB,  draw  the  angle  BAD=ABC,  by 
the  last  problem. 

AD  and  CB  make  the  same  angle  with  AB;  they  are,  therefore, 
parallel.     (Definition  of  parallel  lines). 

PROBLEM    7. 

To  divide  a  given  line  into  any  number  of  equal  parts. 
Let  A  B  represent  the  given  line,  and 
let  it  be  required  to  divide  it  into  any 
number  of  equal  parts,  say  five.    From 
one  end  of  the  line  A,  draw  AD,  inde- 
finite in  both  length  and  position.    Take 
any  convenient  distance  in  the  dividers, 
as  Aa,  and  set  it  off  on  the  line  AD; 
thus  making  the  parts  Aa,  ab,  be,  <kc.,  equal.     Through  the  last 
point,    e,  draw  EB,  and  through  the  points  a,  b,  c,  and  d,  draw 
parallels  to  eB  (problem  6.);  these  parallels  will  divide  the  line  as 
required     Proof  (th.  17,  b.  2). 


INTRODUCTION 


15 


PROBLEM    8. 

To  find  a  third  proportional  to  two  given  lines. 

Let.  AB  and  AC  be  any  two  lines.     Place     A 

them  at  any  angle,  and  join  CB.     On  the     A 
greater  line,  AB,  take  AD— A  C,  and  through 
D,  draw  DE  parallel  to  EG;  AE  is  the  third 
proportional  required. 

Proof,  (th.  17,  b.  2). 


B 


PROBLEM    9. 

To  find  a  fourth  proportional  to  three  given  lines. 

Let  AB,   AC,   AD,   represent    the      A 

three  given  lines.  Place  the  first  two 
together,  at  a  point  forming  any  angle, 
as  BAG,  and  join  BO.  On  AB  place 
AD,  and  from  the  point  D,  draw 
(problem  6)  DE  parallel  to  BO;  AE 
will  be  the  fourth  proportional  required. 

Proof,  (th.  17,  b.  2). 


B 


PROBLEM     10 

To  find  the  middle,  or  mean  proportional, 

Place  AB  and  BC  in  one  right  line, 
and,  on  AC,  as  a  diameter,  describe  a 
semicircle  (postulate  3),  and  from  the 
point  B,  draw  ED  at  right  angles  to  A  0 
(problem  3);  ED  is  the  mean  propor- 
tional required. 

Proof,  (scholium  to  th.  17,  b.  3). 


lines. 


16 


SURVEY1  JNU. 


PROBLEM     11. 

To  find  the  center  of  a  given  circle. 

Draw  any  two  chords  in  the  given  circle, 
as  AB  and  CD;  and  from  the  middle  point, 
»,  of  AB,  draw  a  perpendicular  to  AB; 
and  from  the  middle  point,  m,  draw  a  per- 
pendicular to  CD;  and  where  these  two 
perpendiculars  intersect  will  be  the  center 
of  the  circle.  Proof,  (th.  1,  b.  3). 


PROBLEM     12. 

To  draw  a  tangent  to  a  given  circle,  from  a  given  point,  either  in 
or  without  the  circumference  of  the  circle. 

When  the  given  point  is  in  the  circum- 
ference, as  A,  draw  AC  the  radius,  and 
from  the  point  A,  draw  AB  perpendicular 
to  A  C;  AB  is  the  tangent  required. 

Proof,  (th.  4,  b.  3). 


When  A  is  without  the  circle,  draw 
AC  to  the  center  of  the  circle  ;  and  on 
AC,  as  a  diameter,  describe  a  semi- 
circle ;  and  from  the  point  B,  where 
this  semicircle  intersects  the  given 
circle,  draw  AB,  and  it  will  be  tangent 
to  the  circle. 

Proof,  (th.  9,  b.  3),  and  (th.  4,  b.  3). 


PROBLEM     13. 

On  a  given  line,  to  describe  a  segment  of  a  circle,  thai  shall  contain 
an  angle  equal  to  a  given  angle. 


INTRODUCTION. 


17 


Let  AB  be  the  given  line,  and  C 
the  given  angle.  At  the  ends  of  the 
given  line,  make  angles  DAB,  DBA, 
each  equal  to  the  given  angle,  C. 
Then  draw  AE,  ^^perpendiculars  to 
AD,  ED;  and  from  the  center,  E,  with 
radius,  EA  or  EB,  describe  a  circle ; 
then  AFB  will  be  the  segment  required,  as  any  angle  F,  made  in 
it,  will  be  equal  to  the  given  angle,  C. 

Proof,  (th  11.  b.  3),  and  (th.  8,  b.  3). 


PROBLEM     14. 

To  cut  a  segment  from  any  given  circle,  that  shall  contain  a  given 
angle. 

Let  C  be  the  given  angle.  Take 
any  point,  as  A,  in  the  circumference, 
and  from  that  point  draw  the  tangent 
AB;  and  from  the  point  A,  in  the  line 
AB,  make  the  angle  BAD=C  (pro- 
blem 5),  and  AED  is  the  segment 
required. 

Proof,  (th.  11,  b.  3),  and  (th.  8,  b.  3) 


PROBLEM     15. 

To  construct  an  equilateral  triangle  on  a  given  Jinite  straight  line. 

Let  AB  be  the  given  line,  and  from  one 
extremity,  A,  as  a  center,  with  a  radius 
equal  to  AB,  describe  an  arc.  At  the  other 
extremity,  B,  with  the  same  radius,  describe 
another  arc.  From  C,  where  these  two 
arcs  intersect,  draw  CA  and  CB;  ABC  will 
be  the  triangle  required. 

The  construction  is  asuffici-ent  demonstration.     Or,  (ax.  1). 


18 


SURVEYING. 


D 


PROBLEM     16. 

To  construct  a  triangle,  having  its  three  sides  equal  to  three  given 
lines,  any  two  of  which  shall  be  greater  than  the  third. 

Let  AB,  CD,  and  EF  represent  the  three      E F 

lines.  Take  any  one  of  them,  as  AB,  to  be  one 
side  of  the  triangle.  From  A,  as  a  center,  with 
a  radius  equal  to  CD,  describe  an  arc;  and 
from  B,  as  a  center,  with  a  radius  equal  to  EF, 
describe  another  arc,  cutting  the  former  in  n. 
Join  An  and  Bn,  and  AnB  will  be  the  A 
required.  Proof,  (ax.  1). 


PROBLEM     17. 

To  describe  a  square  on  a  given  line. 

Let  AB  be  the  given  line,  and  from  the  extre- 
mities, A  and  B,  draw  A  C  and  BD  perpendicular 
to  AB.  (Problems.) 

From  A,  as  a  center,  with  AB  as  radius,  strike 
an  arc  across  the  perpendicular  at  C;  and  from  0, 
draw  CD  parallel  to  AB;  ACDB  is  the  square 
required.  Proof,  (th.  21,  b.  1.) 


B 


PROBLEM     18. 

To  construct  a  rectangle,  or  a  parallelogram,  whose  adjacent  sides 
are  equal  to  two  given  lines. 

Let  AB  and  AC  be  the  two  given  lines.      A C 

From  the  extremities  of  one  line,  draw  per-      A B 

pendiculars  to  that  line,  as  in  the  last  problem  ;  and  from  these 
perpendiculars,  cut  off  portions  equal  to  the  other  line ;  and  by  a 
parallel,  complete  the  figure. 

When  the  figure  is  to  be  a  parallelogram,  with  oblique  angles, 
describe  the  angles  by  problem  5.     Proof,  (th.  21,  b.  1). 


INTRODUCTION.  19 

PROBLEM     19. 

To  describe  a  rectangle  that  shall  be  equal  to  a  given  square,  and 
have  a  side  equal  to  a  given  line. 

Let  AB  be  a  side  of  the  given  square,  and      <7 D 

CD  one  side  of  the  required  rectangle.  A B 

Find  the  third  proportional,  EF,  to  CD      E F 

and  AB  (problem  8).     Then  we  shall  have, 
CD  :  AB  :  :  AB  :  EF 

Construct  a  rectangle  with  the  two  given  lines,  CD  and  EF 
(problem  18),  and  it  will  be  equal  to  the  given  square,  (th.  13,  b.  2). 

PROBLEM    20. 

To  construct  a  square  that  shall  be  equal  to  the  difference  of  two 
given  squares. 

Let  A  represent  a  side  of  the  greater  of  two  given  squares,  and 
B  a  side  of  the  lesser  square. 

On  A,  as  a  diameter,  describe  a  semi- 
circle, and  from  one  extremity,  m,  as  a  cen- 
ter, with  a  radius  equal  to  By  describe  an 
arc,  n,  and,  from  the  point  where  it  cuts  the 
circumference,  draw  mn  and  np;  np  is  the 
side  of  a  square,  which,  when  constructed, 
(problem  17),  will  be  equal  to  the  difference 
of  the  two  given  squares.  Proof,  (th.  9,  b.  3,  and  36,  b.  1.) 

PROBLEM    21. 

To  construct  a  square,  that  shall  be  to  a  given  square,  as  a  line,  M, 
to  a  line,  N. 

Place  M  and  N  in  a  line,  and  on  the  sum  describe  a  semicircle, 
From  the  point  where  they  join,  draw  a  perpendicular  to  meet  the 
circumference  in  A.  Join  Am  and 
An,  and  produce  them  indefinitely. 
On  Am  or  An,  produced,  take  AB= 
to  the  side  of  the  given  square  ;  and 
from  B,  draw  BC  parallel  to  mn; 
A  C  is  a  side  of  the  required  square. 


90  SURVEYING. 

Besides  the  numerical  scale  of  equal  parts,  we  hare  scales  of 
chords,  sines,  and  tangents,  which  can  be  constructed  corresponding 
to  any  radius. 

Such  scales  of  course  are  not  scales  of  equal  parts. 

Such  scales  are  constructed  hi  the  following  manner. 

Take  CA  any  radius,  and  describe  a  semicircle.  Draw  CD  at 
right  angles  to  AB,  and  draw  a  tangent  line  from  A.  Divide  the 
arc  AD  into  equal  parts  10,  20,  30,  &c.,  beginning  at  D,  and 
subdivide  them  as  much  as  required. 

Draw  10  10, — | 
20  20,  — 3030,  to.,  I 
all  parallel  to  CD. 

From  C  to  10  on  the  | 
line  CA,  is  the  sine  of 
10°.     From  C  to  20  is  I 
the  sine  of  20°&c.&c.| 

The  line   10  10  is 
the  sine  of  80°,  and  CD  01  CA  is  the  sine  of  90°. 

The  distance  from  A  to  D  is  the  chord  of  90°,  from  A  to  10  is 
the  chord  of  80°,  and  from  A  to  20  is  the  chord  of  70°,  and  so  on 
down.  Thus  we  perceive  that  we  can  take  off  any  sine  or  chord  and 
lay  it  down  on  a  ruler ;  and  chords  and  sines  thus  laid  off  constitute 
the  scale  of  chords,  sines,  <fec. 

Lines  drawn  from  C  through  any  division  of  the  arc,  commencing 
at  A  to  strike  the  tangent  line,  will  mark  off  the  tangent  correspond- 
ing to  that  arc.  Thus,  if  the  angle  ACH  is  30°,  then  the  line  AH 
placed  on  a  scale,  will  represent  the  tangent  of  30°  to  the  radius  CA, 
and  thus  any  other  tangent  can  be  laid  down  on  the  same  scale. 

The  scale  of  chords  and  sines,  as  well  as  the  scale  of  equal  parts, 
are  to  be  found  on  the 

SECTOR. 

The  sector  is  commonly  made  of  ivory,  and  consists  of  two  arms 
which  open  and  turn  reund  a  joint  at  their  common  extremity. 

For  some  operations,  particularly  the  projection  of  solar  eclipses, 
the  sector  is  a  very  useful  instrument. 


INTRODUCTION. 


The  figure  before  us  represents  one  side  of  a  sector  with  the  plane 
scale  only  upon  it.  More  than  one  scale  can  be  put  on  to  a  side, 
but  we  represent  but  one  to  avoid  confusion. 

The  scale  must  be  alike  on  both  arms  —  and  it  must  commence 
exactly  at  the  joint  —  hence  when  near  the  center  the  different  scales 
crowd  each  other. 

The  two  arms  of  the  sector  always  form  two  sides  of  a  triangle, 
and  by  opening  and  closing  them  we  vary  the  angle,  yet  the  distance 
across  from  one  arm  to  the  other  is  always  proportional  to  the  sides 
of  the  triangle. 

The  advantage  of  the  sector  will  appear  from  the  following 
problem. 

A  map  is  before  me,  its  scale  is  20  miles  to  an  inch ;  I  wish  to 
find  the  distance  in  a  right  line  between  two  points  laid  down  on  it. 

1st.  I  take  one  inch  in  the  dividers  and  open  the  sector,  so  that 
the  distance  between  20  and  20  on  the  two  arms,  shall  just  corres- 
pond to  the  measure  in  the  dividers,  that  is,  shall  be  one  inch.  Let 
the  sector  lie  on  the  table  thus  opened. 

2nd.  Now  take  the  distance  you  wish  to  measure,  in  the  dividers  ; 
place  one  foot  on  one  arm  of  the  sector,  and  the  other  foot  on  the 
other  aim;  so  that  the  feet  of  the  dividers  shall  fall  on  the  same 
number  on  both  arms  of  the  sector. 

The  number  thus  marked  by  the  dividers  will  be  the  distance 
required.  The  distance  between  any  other  two  points  may  be 
measured  on  the  same  map,  without  any  computation  whatever. 

For  another  illustration  of  the  utility  of  the  sector,  let  us  suppose, 
that  the  sine  of  20°  is  required  corresponding  to  a  radius  of  6  inches. 


22  SURVEYING. 

Take  6  inches  in  the  dividers,  and  open  the  sector  so  that  the  sine 
of  90°  from  arm  to  arm  shall  be  6  inches. 

The  sector  being  thus  open,  take  the  distance  from  20  to  20,  on 
the  line  of  sines  from  arm  to  arm,  in  the  dividers,  and  that  is  the 
distance  required. 

O  U  N  T  E  R'  S     SCALE. 

Gunter's  scale  is  commonly  two  feet  in  length,  containing  the 
plane  scale  and  the  scale  of  sines,  chords,  and  tangents  on  one  side 
of  it,  and  the  scale  for  the  logarithms  of  numbers,  sines,  and  tangents 
on  the  other. 

This  scale  is  very  ingenious,  but  it  is  not  so  much  used  nor  con- 
sidered so  important  as  formerly. 


CHAP  TE  R    II. 

LOGARITHMS. 

ART  1.  Logarithms  are  exponents. 
Thus,  if  a3  =9 

and  a3 =27 

Then  a5  =243  ;    by  multiplying  the    two 

equations  together  term  by  term. 

The  exponent  2  of  the  first  equation  may  be  considered  the  log- 
arithm of  9  ;  the  exponent  3  the  logarithm  of  27,  and  the  exponent 
5  the  logarithm  of  the  number  243. 

In  these  equations  a=3  the  base  of  the  system. 

By  the  preceding  operation  it  is  obvious  that  adding  the  exponents 
2  and  3,  corresponds  to  multiplying  the  numbers  9  and  27. 

If  we  take  the  equation  a5  =243,  and  divide  it  by  a3 =9  member 
by  member  we  shall  have 

a5~2=a3=27. 

Hence  adding  exponents  (logarithms)  corresponds  to  the  mitltiplica- 


LOGARITHMS.  23 

tion  of  their  corresponding  numbers  and  subtracting  the  exponents 
(logarithms)  corresponds  to  the  division  of  their  numbers. 

It  is  this  property  of  logarithms  that  gives  them  their  utility  and 
importance. 

ART.  2»  The  base  of  our  common  system  of  logarithms  is  10,  and 
in  any  equation  in  the  form  10x=w,  x  is  the  logarithm  of  the  num- 
ber n  whatever  number  n  may  represent.  If  w=10,  then  the 
equation  becomes  10*=  10.  Whence  x—  1  because  101  =  10.  There- 
fore hi  our  common  system  of  logarithms  the  logarithm  of  10  must 
bel. 

Now  because  1  0  °  =  1 

101  =  10 
10'  =  100 
103  =  1000 
10*  =  10000 

&c.       &c.  ;    it  is  plain  that  the  log- 

arithm of   1  is  0,  of  10  is   1,  of  100  is  2,  <kc.,  every  power  of  10 
increasing  the  logarithm  by  1  . 

It  is  also  obvious,  that  every  number  between  1  and  10  must  have 
a  fractional  or  decimal  number  for  its  logarithm,  and  every  number 
between  10  and  100  must  have  owe,  and  some  decimal  for  iU 
logarithm. 

In  the  equation  10*=  3,  x  is  the  logarithm  of  3,  and  if  we 
multiply  this  by  101=10,  member  by  member,  we  shall  have 

10l+*=30 

Multiply  this  by  1Q1     =10 

and  we  have 


These  results  show  that  3,  30,  300,  have  logarithms  containing 
the  same  decimal  number  ;  x  differs  from  each  exponent  only 
by  whole  numbers,  and  thus,  generally  ;  any  number  multiplied  or 
divided  by  10,  or  any  power  of  10,  will  have  logarithms  containing 
the  same  decimal  part. 

ART.  3*  For  the  general  computation  of  logarithms  we  refer  to 
algebra,  and  in  a  work  like  this  we  shall  only  attend  to  such  portions 
of  theory  as  to  enable  the  student  to  use  them  understandingly  and 
with  as  much  practical  facility  as  possible. 


24  SURVEYING. 

Let  it  be  observed  that  the  logarithm 

of     10000  is     4,00000 

of      1000  is     3,00000 

of         100  is     2,00000 

of          10  is     1,00000 

of            1  is     0,00000 

Ti_  is— 1,00000 

Ti__= ID-*  is— 2,00000 

For  every  division  of  the  number  by  10  we  subtract  1  from  its 
logarithm,  and  when  the  number  comes  down  to  1,  and  its  logarithm 
of  course  to  0,  if  we  again  divide  by  10,  making  it  Jv  or  10-1,  we 
must  subtract  one  from  the  logarithm,  making  it  — 1. 

The  decimal  portion  of  a  logarithm  is  always  positive,  but  the  index 
or  whole  number  part  of  it,  becomes  minus  w/ten  the  value  of  the 
number  is  less  than  1 . 

ART.  4.  The  whole  number  belonging  to  any  logarithm  is  called 
its  index,  a  very  appropriate  term,  because  it  indicates,  it  points  out 
where  to  place  the  decimal  point  between  whole  numbers  and 
decimals. 

The  index,  or  (as  some  call  it)  the  characteristic,  is  never  put  in 
the  tables  (except  from  1  to  100),  because  we  always  know  what 
it  is.  It  is  always  one  less  than  the  number  of  digits  in  the  whole 
number.  This  is  obvious  from  Art.  3. 

Thus,  the  number  3754  has  3  for  the  index  of  its  logarithm, 
because  the  number  consists  of  4  digits  ;  that  is,  the  logarithm  is  3, 
and  some  decimal. 

The  number  34.785  has  1  for  the  index  of  its  logarithm,  because 
the  number  is  between  34  and  35,  and  1  is  the  index  for  all  num- 
bers between  1  and  100. 

All  numbers  consisting  of  the  same  figures,  whether  integral, 
fractional,  or  mixed,  have  logarithms  consisting  of  the  same  decimal 
part.  (Art.  2.)  The  logarithms  differ  only  in  their  indices. 

Thus,  the  number  7956.  has  3.900695  for  its  log. 

the  number  795.6  has  2.900695         " 

the  number  79.56  has  1.900695 

the  number  7.956  has  0.900695        « 


LOGARITHMS.  25 

the  number    .7956    has  — 1.900695  for  its  log. 
the  number  .07956     has  —2.900695         " 

&c.,  &c. 

For  every  division  by  10,  we  diminish  the  index  by  1.  When 
the  index  is  minus  it  indicates  a  decimal  number ;  but  let  the  learner 
remember  that  the  index  only  is  minus  ;  the  decimal  part  is  always 
positive. 

ART.  5*  To  take  out  the  logarithm  of  any  number  from  the 
tables,  we  only  consider  the  digits ;  for  the  logarithms  of  7956,  or 
of  7.956,  or  of  .007956,  have  the  same  decimal  part;  and  when 
that  decimal  part  is  found  we  then  consider  the  value  of  the  number 
to  prefix  the  index. 

To  prefix  the  index  to  a  decimal,  count  the  decimal  point  1,  and 
each  cipher  1,  up  to  the  first  significant  figure,  and  this  is  the  neg- 
ative index. 

For  example,  find  the  logarithm  of  the  decimal  .00085.  To 
accomplish  this  we  must  look  for  the  logarithm  of  the  whole  num- 
ber 85,  and  we  find  its  decimal  part  to  be  .929419 ;  and  now,  to 
determine  the  index,  we  count  one  for  the  decimal  point  and  three 
ciphers,  making  4  ;  hence,  we  have  Num.  .00085  -  log.  — 4.929419. 

The  smaller  the  decimal,  the  greater  the  negative  index;  and 
when  the  decimal  becomes  0,  the  logarithm  becomes  negatively 
infinite. 

ART.  6.  The  logarithm  of  any  number  consisting  of  four  digits 
or  less,  can  be  taken  out  of  the  table  directly  and  without  the  least 
difficulty. 

Thus,  to  find  the  logarithm  of  the  number  3725,  we  find  the 
number  372  at  the  side,  and  over  the  top  we  find  5,  and  opposite 
the  former  and  under  the  latter  we  find  .571126  for  the  decimal 
part  of  the  logarithm.  The  57,  the  first  two  decimal,  is  under  0, 
which  is  the  same  for  the  whole  horizontal  column. 

Hence,  the  logarithm  of  3725  is  3.571126 
of  37250  is  4.571126 
of  3.725  is  0.571126 

&c.,  <fec. 

Find  the  logarithm  of  1176.  We  find  117  at  the  side,  and  6  at 
the  top,  and  opposite  the  former  and  under  the  latter  we  find  .407 


20  SURVEYING. 

The  point  here  demands  a  cipher,  and  is  put  in  to  arrest  attention 
to  make  the  operator  look  to  the  next  horizontal  line  below  for  the 
first  two  decimals.  Thus,  we  find  .070407  for  the  decimal  part 
of  the  logarithm  required. 

Hence,  the  log.  of  1176  is  3.070407 

1.  What  is  the  log.  of  .001176  ?  Ans.  —3.070407 

2.  What  is  the  log.  of       13.81?  Ans.       1.140194 

3.  What  is  the  log.  of       72.55  ?  Ans.       1.860637 

4.  What  is  the  log.  of       .6762  ?  Ans.  —1.830075 

5.  What  is  the  logarithm  of  the  number  834785  ? 

This  number  is  so  large  that  we  cannot  find  it  in  the  table,  but 
we  can  find  the  numbers  8347  and  8348.  The  logarithms  of  these 
numbers  are  the  same  as  the  logarithms  of  the  numbers  834700 
and  834800,  except  the  indices. 

834700     log.     5.921530 
834800     log.     5.921582 

Difference,  100  ~~52 

Now,  our  proposed  number,  834785,  is  between  the  two  preced- 
ing numbers;  and,  of  course,  its  logarithm  lies  between  the  two 
preceding  logarithms ;  and,  without  further  comment,  we  may  pro- 
portion to  it  thus,  100  :  85=52  :  44.2 

Or,  1.  :. 85=52  :  44.2 

To  the  logarithm  5.921530 

Add  44 

Hence,  the  logarithm  of    834785    is  5.921574 

the  logarithm  of  83.4785    is  1.921574 

From  this  we  draw  the  following  rule  to  find  the  logarithm  of 
any  number  consisting  of  more  than  four  places  of  figures. 

RULE. — Take  out  the  logarithm  of  the  four  superior  places  directly 
from  the  table,  and  take  the  difference  between  this  logarithm  and  the 
next  greater  logarithm  in  tlie  table.  Multiply  this  difference  by  the 
inferior  places  in  the  number  as  a  decimal,  and  add  the  result  to  the 
logarithm  corresponding  to  the  superior  places^  the  sum  will  be  the 
logarithm  required. 

Example.     Find  the  log.  of   357.32514. 

The  four  superior  digits   are   3573;    the  logarithm  of  these 


LOGARITHMS.  27 

corresponds  to  the  decimal,  .553033,  for  its  decimal  part.     The 
inferior  digits,  taken  as  a  decimal,  are 

.2514 

122 

5028 
5028 
2514 
30.6708 

This  result  shows  that  30,  or  more  nearly,  31,  should  be  added 
to  the  logarithm  already  found,  thus  giving  .553064  for  the  deci- 
mal part  of  the  logarithm  357.32514. 

Therefore,  as  three  digits  of  the  given  number  are  whole  num- 
bers, the  index  must  be  2,  and  the  logarithm 

of    357.32514    is      2.553064 

of    3573251.4    is      6.553064 

of  .035732514    is  —2.553064 

The  change  between  the  place  of  the  decimal  point  in  a  number, 
and  the  corresponding  change  of  the  index  to  its  logarithm,  should 
be  strongly  impressed  on  the  mind  of  a  learner. 

Example  2.  What  is  the  log.  of   366.25636  ?     Ans.  2.563785 

3.  What  is  the  log.  of     39.37079  ?     Ans.  1.595174 

4.  What  is  the  log.  of       2.37581  ?     Ans.  0.375812 

ART.  7.  We  now  give  the  converse  of  the  last  article ;  that  is, 
we  give  the  decimal  part  of  a  logarithm  to  find  its  corresponding 
number. 

Taking  the  decimal  hi  Example  1,  (Art.  6,)  .553064,  we  demand 
its  corresponding  number.* 

The  next  less  logarithm  in  the  table,  is  .553033,  corresponding  to 
the  figure  3573.  The  difference  between  this  given  logarithm  and 
the  one  next  less  hi  the  table,  is  31  ;  and  the  difference  between  two 
consecutive  logarithms  in  this  part  of  the  table,  is  122.  Now  divide 
31  by  122,  and  write  the  quotient  after  the  number  3573. 

*  To  take  out  a  number  from  its  logarithm,  never  enter  the  first  part  of  tha 
table  between  1  and  100.  Go  to  the  main  table,  as  it  contains  many  more 
logarithms. 


*  SURVEYING 

That  is,  122)31.(254 

244 


660 
610 

500 
488 

The  figures,  then,  are  3573254,  which  corresponds  to  the  decimal 
logarithm  .553064;  and  the  value  of  these  figures  will,  of  course, 
depend  on  the  index  to  the  logarithm. 

If  this  given  logarithm  contained  an  index,  such  index  would 
point  out  how  many  of  these  figures  must  be  taken  for  whole  num- 
bers, the  others  will  be  decimals  ;  thus,  if  the  index  had  been  4,  the 
number  would  be  35732.54 

If  the  given  decimal  had  been  .553063.67,  which  is  the  exact 
converse  of  example  1,  then  we  should  have  found  that  number, 
35732514 ;  but  we  did  not  give  that  decimal  logarithm,  because  the 
table  contains  only  six  decimal  places.  From  this  obvious  operation 
we  derive  the  following  rule  to  find  the  number  corresponding  to  a 
given  logarithm. 

RULE.  —  If  the  given  logarithm  is  not  in  the  table,  find  the  one  next 
less,  and  take  out  the  four  figures  corresponding  ;  and  if  more  them 
four  figures  are  required,  take  the  differenee  "between  the  given  logarithm 
and  the  next  less  in,  the  table,  and  divide  that  difference  by  the  difference 
of  the  two  consecutive  logarithms  in  the  table,  the  one  less,  the  other 
greater  than  the  given  logarithm  ;  and  the  figures  arising  in  the  quotient, 
as  many  as  may  be  required,  must  be  annexed  to  the  former  figures 
taken  from  the  table. 

EXAMPLES. 

1.  Given,  the  logarithm  3.743210,  to  find  its  corresponding  num- 
ber true  to  three  places  of  decimals.  Ans.  5536.182 

2.  Given,  the   logarithm    2.633356,   to  find   its   corresponding 
number  true  to  two  places  of  decimals.  Ans.  429.89 

3.  Given,  the  logarithm  — 3.291742,  to  find   its  corresponding 
number.  Ans.  .0019577 


LOGARITHMS.  20 

MULTIPLICATION    BY    LOGARITHMS. 

ART.  8.  If  the  principle  first  laid  down  in  (ART.  1)  is  true,  the 
sum  of  the  exponents  will  be  the  exponent  of  the  product  of  any 
number  of  factors.  In  other  words, 

The  sum  of  the  logarithms  of  any  number  of  factors  will  be  the 
logarithm  of  the  product  of  those  factors. 

N.  B.  The  logarithmic  table  corresponds  to  this  principle,  and  we 
may  see  by  the  following 

EXAMPLES. 

The  log.  of  3  (taken  from  the  table,)  is    0.477121 
The  log.  of  4      "         "     "       "        is    0.602060 

Therefore  the  log.  of  12  must  be  1.079181 

Given,  the  log.  of  7  and  the  log.  of  9,  to  find  the  logarithm  of  63. 
Because  7X9=63,  therefore, 

To  log.     7=0.845098 

Add  log.  9=0.954243 

Sum  1.799341 

By  inspecting  the  table,  we  shall  find  this  logarithm  stands 
opposite  63,  and  by  this  process  the  logarithms  of  all  the  composite 
numbers  have  been  found.  In  this  we  may  consider  that  the  log- 
arithm pointed  out  the  product  63. 

Hence  we  have  the  following  rule  for  obtaining  the  product  of  any 
number  of  factors. 

RULE.  —  Find  the  logarithm  of  each  factor,  add  those  logarithms 
together  and  the  sum  mil  be  the  logarithm  of  the  product.  The  number 
corresponding  to  this  last  logarithm  taken  from  the  tablet  will  be  the 
product  itself. 

EXAMPLES. 


1.  To  multiply  23.14  by  5.062. 

Numbers.  Logs. 

23.14       1.364363 
6.062       0.704322 

Product  117.1347       2.068685 


2.  To    multiply   2.581926    by 
3.457291. 

Numbers.  Logs. 

2.581926   0.411944 
3.457291  0.538736 

Product.  8.92648  0.950680 


30 


SURVEYING. 


3.  To  mult.  3.902  and  597.16 
and  .0314728  all  together. 

Numbers.  Logs. 

3.902 
597.16 
.0314728 
Prod.    73.3333 


0.591287 

2.776091 

—2.497935 


1.865313 

Here  the — 2  cancels  the  2,  and 
the  one  to  carry  from  the  decimals 
is  set  down. 


4.  To  mult.  3.586,  and  2.1046, 
and  0.8372,  and  0.0294  all 
together. 

Numbers.  Logs. 

3.586  0.554610 

2.1046  0.323170 

0.8372  —1.922829 

0.0294  —2.468347 

Prod.  0.1057618    —1.268956 


Here  the  2  to  carry  cancels  the 
— 2,  and  there  remains  the — 1  to 
set  down. 


DIVISION    BY    LOGARITHMS. 

ART.  9,  As  division  is  the  converse  of  multiplication  we  draw  the 
following  rule  for  division  by  use  of  logarithms. 

N.  B.  Addition  and  subtraction  is  to  be  understood  in  the  algebraic 


sense. 


BULB.  —  Front  the  logarithm  of  the  dividend  subtract  the  logarithm 
of  the  divisor,  and  the  number  corresponding  to  the  remainder  is  the 
quotient  required. 


EXAMPLES. 

1.  Divide  327.5  by  2207 

log.     327.5 
log.     2207 
.14839 


Quotient 
2.  Divide  .054  by  1.75 

log.    .054 
log.    1.75 

Quotient 


2.515211 

3.342028 

— M73183 


—•2.732394 
0.243038 
.030857  —2.489356 

ART.  10.  The  preceding  examples  in  multiplication  and  division 
were  adduced  only  to  show  the  nature  of  logarithms :  had  our 
object  been  results,  the  common  arithmetical  operations  would  have 
been  more  convenient  for  some  of  them ;  but  there  are  cases  that 
demand  the  use  of  logarithms,  and  such  cases  mostly  occur  hi 
Involution  and  Evolution. 


LOGARITHMS. 


31 


RULE  FOR  INVOLUTION. — Take  out  the  logarithm  of  the  given 
number,  and  multiply  it  by  the  index  of  the  proposed  power.  Find 
the  number  corresponding  to  the  product,  and  it  will  be  the  power 
required. 

EXAMPLES. 


1.  What  is  the  2d  power  of 
351? 

log.        351         2.545307 
2 

Ans.  123201         6.090614 

3.  What  is  the  4th  power  of 
.0916  ? 

log.     .0916     —2.961895 
4 

Ans.     .000070401     —5.847580 

Here  4  tunes  the  negative  in- 
dex is  — 8,  adding  the  3  to  carry 
gives  — 5. 


2.  What  is  the  cube  of  1.72  ? 
log.       1.72         0.235528 
3 


Ans. 


5.0884        0.706584 


4.  What  is  the  17th  power  of 
1.04? 

log.       1.04        0.017033 
17 

0.119231 
0.17033 


Ans.  1.9476         0.289561 

N.  B.  This  last  example  be- 
gins to  disclose  the  utility  of  log- 
arithms. 


6.  What  is  the  6th  power  of 
1.037? 

log.     1.037         0.015779 
6 


Ans. 


1.243-1-     0.094674 


6.  What  is  the  21st  power  of 
2.02? 

log.       2.02         0.305351 
21 

.305351 
6.10702 

Ans.       2584454.6        6.412371 


EVOLUTION 

ART.  lit  Evolution  is  the  converse  of  Involution;  hence  we 
have  the  following  rule  for  the  extraction  of  roots  : 

Take  the  logarithm  of  the  given  number  out  of  the  table.  Divide 
the  logarithm,  thus  found,  by  the  index  of  the  required  root ;  then  the 
number  corresponding  is  the  root  sought. 


32  SURVEYING. 


EXAMPLES. 


1.  What  is  the  cube  root  of 


125? 


log.  125          3)2.096910 


Ans.             5  0.698970 

3.  What  is  the  4th  root  of 
751? 

log.  751  4)2.875640 

Ans.          5.235-t-  0.718910 


2.  What  is  the  cube  root  of 


200? 

log.  200          3)2.301030 

Ans.          5.848-i-          0.767010 

4.  What  is  the  20th  root  of 
1.035? 

log.  1.035     20)0.014940 

Ans.        1.001718  0.000747 

5.  What  is  the  cube  root  of  the  decimal     .00048 
log.     .00048          —4.681241 

To  the  inexperienced  here  would  be  a  difficulty,  as  the  index  is 
negative,  and  the  decimal  part  positive.  How  then  shall  we  divide 
by  3  ?  Add  — 2  and  -f-2  to  the  index ;  and  this  is,  in  effect, 
adding  nothing ;  it  merely  changes  the  form  of  the  index,  thus, 
— 6 -f- 2.6  8 1241  Now,  we  can  divide  by  3,  and  the  quotient  is 
— 2.893747.  The  corresponding  number,  or  root,  sought,  is 
.07829+  Ans. 

REMARK. — In  the  preceding  articles  we  have  taught  all  the  preliminary  rules 
for  the  use  of  logarithms  ;  "but  there  is  a  wisdom  beyond  rules,"  and  he  who 
does  not  arrive  at  it,  attains  only  the  burdens  of  knowledge  without  its  bene- 
fits. Rules  are  necessary  through  the  first  rudiments  of  any  science  ;  but  he 
who  can  instantly  fall  back  on  to  first  principles,  and  do  the  most  advantageous 
thing  at  the  most  advantageous  point  of  time,  has  a  practical  tact  of  the 
highest  value. 

ART.  12.  What  follows  will  refer  to  no  particular  rules,  but 
will  be  embraced  under  general  principles,  which  are  far  above 
rules. 

To  understand  logarithms  well,  it  is  indispensable  to  study 
the  table,  and  observe  the  increase  of  the  logarithms,  and  com- 
pare that  increase  with  the  increase  of  the  numbers.  For  in- 
stance, the  logarithms  from  1  to  10,  increase  by  1,  and  we  must 
go  ten  times  as  far,  that  is,  to  100,  to  obtain  another  increase  of 
1  in  the  logarithms. 

Hence,  logarithms  to  numbers  near  to  unity,  increase  very 
fast,  and  far  from  unity,  increase  very  slowly  ;  and  this  is  true 


LOGARITHMS.  33 

whichever  side  of  unity  the  number  may  be,  above  or  below. 
For  instance,  4  is  as  much  above  unity  as  £  is  below  unity  ;  so 
623  is  as  much  above  unity  as  ^fg  is  below  unity. 

Now  623  multiplied  by  ^£3  will  .produce  1,  therefore  the  log. 
of  623  added  to  the  log.  of  e^-3,  its  reciprocal,  must  produce  the 
log.  of  1,  or  zero. 

Observe  the  decimal  logarithms  on  page  3,  and  compare  them 
with  the  logarithms  on  page  20.  Those  on  page  3  are  small  in 
value,  and  vary  rapidly;  those  on  page  20  are  large  in  value,  and 
vary  very  slowly. 

Hence  it  is,  practically,  more  difficult  to  adjust  a  log.  to  its 
number,  when  the  decimal  part  of  a  logarithm  is  small,  than 
when  it  is  large  ;  but  we  can  avoid  the  use  of  a  small  decimal 
in  a  logarithm,  as  the  following  artifice  will  show. 

Let  us  take  Example  4,  Art.  11.  That  is,  find  the  number 
corresponding  to  the  log.  0.000747 

Subtract  the  log.  of  1.01  0.004321     (a) 

The  log.  of    .9918  is  —1.996426     (b) 

The  two  logarithms  (a)  and  (b)  added  together,  will  produce 
(0.000747)  the  given  logarithm,  and  the  number  corresponding 
to  this  log.  will  be  the  product  of  the  two  numbers  1.01,  0.9918, 
or  1.001718. 

Our  given  log.  (0.000747)  has  a  small  decimal  part,  and  sub- 
tracting the  log.  of  1.01,  produced  a  logarithm  having  a  large 
decimal  part,  and  this  was  our  object.  The  large  decimal  loga- 
rithm we  can  carry  to  the  table,  and  take  out  the  corresponding 
number,  to  great  accuracy,  by  mere  inspection. 

ART.  13.  Our  table  only  extends  to  four  figures,  three  at  the 
side,  and  one  at  the  top  of  the  pages,  but  by  a  little  artifice,  the 
table  will  serve  for  any  number. 

For  instance,  suppose  that  the  log.  of  101248  was  required, 
the  uninstructed  might  infer  that  it  could  not  be  obtained, 
because  the  table  does  not  extend  so  far;  but  by  factoring  it,  as 
follows,  we  find  that  it  is  the  product  of  the  two  factors,  16 

and  6328. 

3 


34  SURVEYING. 

2)101248 


6328 

Log.  of  16  is  1.204120 

Log.  of  6328  is  3.801266 

Log.  of  101248  therefore  is  6.005386 

We  may  take  another  artifice  (which  is  the  common  one)  to 
bring  this  number  within  the  scope  of  the  table.  Conceive  it 
divided  by  1000.  Then  the  quotient  would  be  101.248,  and 
this  number  is  between  101  and  102,  and  of  course  within  the 
scope  of  the  table,  but  the  first  method  is  the  most  accurate, 
and  therefore  the  best. 

If  we  demanded  the  log.  of  101249,  a  number  greater  by 
unity  than  the  preceding  number,  which  may  be  a  prime  num- 
ber, and  not  capable  of  being  separated  into  factors,  we  can 
obtain  the  log.  of  101248  as  before,  and  then  add  the  quotient 
arising  from  the  following  division  : 
0.43429448 

y 

The  numerator,  or  dividend,  is  the  modulus  of  our  system, 
and  N  is  the  number  immediately  preceding  the  one  whose  log. 
is  sought.  For  this  example  JV=101248,  and 


101248 

We  carry  the  division  only  to  the  sixth  decimal  place,  as  this 
is  the  limit  of  the  table  now  under  consideration. 

Hence,  the  logarithm  of  the  number  101249  is  5.005390. 

The  expression   J  --  is  the  expression  or  value  of  the 

correction  corresponding  to  one  of  number,  but  if  we  wished  to 
make  a  correction  2.3,  we  would  multiply  the  correction  for  one 


LOGARITHMS.  35 

(.000004)  by  2.3,  and  this  will  give  .000009  for  the  correction 
corresponding  to  an  increase  of  2.3,  thus: 

The  log.  of  101248         is         5.005386 

Add  2.3     add  9 


The  log.  of  101250.3      is         5.005395 

The  log.  of  8091  is  3.908002.     What  is  the  log.  of  8092 
without  using  the  table? 

8091)0.43429448(0.000054  nearly. 
40455 

29744 

To  log.  of  8091,  which  is  3.908002 

Add  54 


Log.  of  8092  is  3.908056 

If  the  correction  had  been  for  part  of  a  unit,  we  should  have 
added  the  like  part  of  .000054,  and  this  is  the  principle,  and  all 
the  principle,  of  deducing  one  log.  from  another. 
Given  the  log.  of  5280  to  find  that  of  52815. 
The  last  number  we  shall  use  as  5281.5,  then  it  will  differ 
from  the  given  number  by  1£. 

5280).43429448(0.000082     correction  for  1. 
42240  41  "  |. 

11894         .000123 
Log.  of  5280  is  3.722634 

Log.  of  52815  is  4.722757 

If  we  factor  the  number  52815,  we  shall  find  that  it  may  be 
regarded  as  the  product  of  two  factors,  3521  and  15. 
Log.  of  3521  is  3.546666 

Log.  of  15  is  1.176091 


Hence,  Log.  of  52815  is  4.722757  as  before. 

For  another  example  :  The  ratio  between  the  diameter  and 
circumference  of  a  circle  is  3.14159265,  find  the  logarithm  of 
this  number.  As  the  number  is  less  than  10,  the  index  will  be 
0 ;  but  to  get  the  logarithm  to  proper  accuracy,  we  will  con- 


36  SURVEYING. 

ceive  the  number  to  be  a  whole  number  as  far  as  31415,  and 
then  we  can  call  the  rest  .9265,  a  decimal.     The  whole  number 
31415,  is  composed  of  the  two  factors,  5  and  6283.     Therefore 
To  the  log.  of  6283  3.798167 

Add  the  log.  of  5  .698970 

Log.  of  3.1415  is  0.497137 

31415).43429448(0.000014X.92     is      13  nearly. 
31415 

120144 

Whence,  the  log.  of  3.141592  is  0.497150, 

as  near  as  our  table  of  six  decimal  places  can  express  it. 

N.  B.  We  have  formed  a  table  consisting  of  twelve  decimal 
places,  (without  any  indices),  which  can  be  used  for  very 
delicate  cases,  and  the  proper  explanations  are  made  in  the 
Appendix. 

ART.  14.  A  person  who  properly  understands  the  principles 
of  logarithms,  can  find  the  log.  of  any  number,  without  a  table, 
provided  he  retains  in  his  memory  the  value  of  the  modulus 
(.43429448),  and  the  value  of  the  log.  of  2,  and  of  3,  and  of  7, 

and  of  11. 

The  log.  of  2  is  0.301030 
The  log.  of  3  is  0.477121 
The  log.  of  7  is  0.845098 
The  log.  of  11  is  1.041393 

The  log.  of  2  has  the  same  decimal  part  as  20,  200,  or  .2 
tenths  ;  the  difference  is  in  the  indices. 

The  same  remark  will  apply  to  3,  and  all  other  integers. 

EXAMPLES. 

1.  Find  the  log.  of  112,  without  the  table,  the  logarithms  of  2 
and  7  being  given. 

112=4.28=4.4.7=2.2.2.2.7. 

Four  times  the  log.  of  2,  plus  the  log.  of  7,  will  be  the  result. 
Log.  of  2  =     0.301030X4=1.204120 
Log.  of  7  845098 

Log.  of  112  2.049218  Ans, 


LOGARITHMS.  37 

2.     Find  the  log.  of  1121,  without  the  me  of  the  table. 

The  log.  of  1120  is  given  in  the  first  example,  and  this  cor- 
rected for  the  increase  of  unity,  will  be  the  result. 

1 120).43429448(0.000388. 
To  3.049218 

Add  .000388 


And  log.  of  1121     is  3.049606    Ans. 

3.     Find  the  log.  of  99  without  the  use  of  the  table. 

99=3all  0.477121 

2 


Log.  of  9  is  0.954242 

Add  log.  of  11  1.041393 

Whence  log.  of  99  is  1.995635    Ans. 

4.  There  are  5280  feet  in  a  mile.     Find  the  log.  of  this  number 
without  the  use  of  the  table. 

5280=243.11.10. 

Whence  4  times  the  log.  of  2  is  (Ex.  1)         1.204120 
Log.  of  3  0.477121 

Log.  of  11. 1C  2.041393 

Whence  log.  of  5280     is  3.722634  Ans. 

5.  The  sidereal  year  consists  of  365.25637  mean  solar  days. 
Find  the  log.  of  this  number  true  to  6  places  of  decimals,  without 
the  use  of  the  table. 

N.  B.  As  the  number  is  between  365  and  366,  the  index  will 
be  2.  Hence,  we  shall  pay  no  attention  to  the  index ;  the  arti- 
fice is  to  obtain  the  true  decimal  part. 

Because  33.11=363=3.11.11. 


38  SURVEYING. 

Twice  the  log.  of  11     is  2.082786 

Log.  of  3  0.477121 

Log.  of  363  therefore  is  2.559907 

We  must  now  correct  this  log.  for  2 
units,  and  the  decimal  .25637,  or  2^ 
units  nearly. 

Correction  for  the  first  unit  is  :.43429448=0.001199 


For  the  second  unit  it  is 


And  for  the  fraction 


363 

.43429448 

364 


=0.001193 


25637=0.000303 
365      / 


Log.  of  365.25637  therefore  is     2.562602   Ans. 

6.  The  diameter  of  the  earth  is  7912  miles.  Find  the  log*  of 
this  number  without  using  the  table. 

7912=2.2.2.989.     But  989=990—1. 

The  log.  of  99  or  990  was  found  in  Ex.  3.  Using  that  result, 
we  have 

Log.  990  2.995635 

Correction  for  1  ,  subtract  439 

Log.  of  989  is  2.995196 

Add  3  times  log.  2,  or  log.  8,  .903090 

Log.  of  7912  therefore  is  3.898286  Ans. 

We  think  that  we  have  fully  illustrated  that  logarithms  can 
be  readily  found,  independently  of  a  table. 

ART.  1  5.  The  converse  of  the  last  problem  is  not  so  obvious. 
When  the  number  is  given,  the  logarithm  of  that  number  can 
be  found,  as  we  have  just  shown  ;  the  converse  of  that  problem 
is  to  find  the  number  corresponding  to  a  given  logarithm.  We 
must  remember  the  log.  of  2,  of  3,  of  7,  and  of  11,  and  also 
know  the  modulus,  but  it  is  better  to  illustrate  by 

EXAMPLES. 

1.     Find  the  number  corresponding  to  the  log.  3.470263. 
1st.  The  log.  of  2  is     .301030  "]       From  these  we  can  find 
The  log.  of  3  is     .477121    \-  various    other    logarithms, 
The  log.  of  7  is     .845098  j  but  we  observe  that  the  de- 


LOGARITHMS.  39 

cimal  part  of  our  given  log.  is  very  nearly  the  log.  of  3,  but  the 
index  is  3,  therefore  the  number  sought  must  consist  of  four 
places,  the  highest  figure  not  quite  3  ;  that  is,  the  number  sought 
is  not  quite  3000.  It  is  probably  not  far  from  2900,  or  not  far 
from  the  double  of  1456,  which  is  the  double  of  729,  which  is 
the  cube  of  9. 

Therefore         4.93=2916.     But  9=38,  or  93=38. 
Twice  the  log.  of  2,  is  log.  4,  0.602060 

6  times  log.  of  3  is  log.  of  729  =  2.862726 

Log.  of  2916,  therefore  is  3.464786 

Which  subtract  from  3.470263 


0.005477 

If  we  can  find  the  number  corresponding  to  this  last  logarithm, 
the  product  of  that  number  into  2916  will  be  the  number  sought. 
We  can  obtain  this,  approximately,  by  means  of  the  modulus. 
Thus  ^43489448^  ft       ftnd          0.005477^ 

2916  Q 

And  2916-)-^=  the  number  sought. 

In  this  case  0=0.000149.        °'0054— =36 . 8  nearly. 

0.000149 

To  2916  add  36.8,  and  we  have  2952.8  nearly,  for  the  number 
required.  But  the  true  number  is  2953,  the  error  of  about  two- 
tenths  arises  from  the  imperfection  of  this  last  operation. 

To  show  that  there  are  several  methods  of  solving  these 
problems  on  the  same  general  principle,  we  will  take  the  same 
problem  as  before. 

The  given  log.  is  3.470263 

The  log.  of  30,  is  1.477121 

The  log.  of  the  unknown  factor  is         1.993142 
The  log.  of  100  is  2,  hence  this  logarithm  under  considera- 
tion must  correspond  to  some  number  near  98.     Now  we  will 
find  the  log.  of  98  to  compare  it  with  this  log. 

98=2. 49=2. 72. 

Log.  of  98=log.  2-f  2  log.  7.=  1.991226 

This  subtracted  from  1.993142 


And  we  have  left  0.001916 


40  SURVEYING. 

Hence,  the  number  sought  is  30.98=2940  plus,  the  correction 
which  is  found  by  the  following  process : 

^1^=0.000148  nearly.  J«!l£l?=,3  nearly. 

2940  .000148 

Therefore,     2940+13=2953,     the  true  number  sought. 

ART.  16.  Let  the  pupil  observe  that  in  the  last  problem  the 
given  log.  (3.470263)  was  diminished  by  the  log.  of  30,  and  the 
remainder  by  the  log.  of  2940,  hence,  the  remaining  log. 
(0.001916)  is  the  difference  between  the  given  logarithm  and 
the  logarithm  of  2940,  and  in  this  sense  it  is  a  differential  of  a 
logarithm,  and  not  a  logarithm  independent  by  itself. 

Considered  as  an  independent  logarithm,  it  does  not  corres- 
pond to  13,  but  to  some  number  a  little  greater  than  unity. 

Let  us  take  it  as  an  independent  logarithm,  and  how  then 
shall  we  find  the  number  corresponding  ? 

The  log.  of  1  is  0.00000,  and  therefore  the  log.  0.001916  may 
be  considered  as  the  difference  between  itself  and  the  log.  of  the 
number  1. 

Hence,  the  correction  to  the  number  1  may  be  found  by  the 
same  process  as  we  have  just  found  the  correction  to  the  num- 
ber 2940. 

The  operation  is  thus : 

.434294,  0.001916_nnn/14j9 

1  0.43429 

Whence  the  log.  of  1.004412  is  0.001916. 

The  product  of  the  three  factors  30. 85. (1.00441 2)  is  2953 
very  nearly,  as  it  ought  to  be. 

From  the  above  operation,  we  can  draw  the  following  rule  to 
find  the  corresponding  number  to  any  very  small  logarithm : 

BULK. — Divide  the  given  logarithm  by  the  modulus,  and  to  the 
quotient  and  unity.  The  sum  will  be  the  number  sought.* 

*  "We  have  drawn  these  rules  from  mere  practical  observations,  without 
any  pretensions  to  science  ;  but  nevertheless,  they  conform  to  strict  analyti- 
cal deductions,  as  found  in  the  differential  calculus. 

In  that  branch  of  science  we  shall  find  the  equation  dx=* — —  in  which 
m  represents  the  modulus  of  a  system  of  logarithms,  darthe  differential  value 


LOGARITHMS.  41 

EXAMPLE.  The  log.  of  a  certain  number  is  0.003296,  find  the 
corresponding  number. 

By  the  rule    °-003296  =0.007589.  Ans.  1.007589. 

.43429 

For  another  example.  The  log.  of  some  number  just  above  unify 
is  0.000123.  What  is  that  number? 

By  the  rule    a0001—  =0.0002832.  Ans.  1.0002832. 

.43429 

We  will  give  but  one  more  example,  independent  of  the  tables, 
and  the  object  of  that  one  is  to  show  the  utility  of  logarithms, 
notwithstanding  we  may  be  under  the  necessity  of  computing 
the  logarithms  expressly  for  the  example,  which  is  the  following: 

Find  the  value  of  the  fifth  power  of  8,  multiplied  by  the  third 
root  of  7,  and  that  product  divided  by  the  fifth  root  of  6. 

By  logarithms  it  will  stand  thus  : 

5Xlog.  8+£  log.  7—i  log.  6=log.a?. 

The  log.  of  8=3  times  the  log.  of  2,  which  is  .903090. 
5Xlog.8  -         -  4.515450 

}log.7         -  -         -      0.281699 

4.797149 
jlog.  6  -  -  0.155630 

Log.  x  -  -     4.641519 

We  must  now  find  the  value  of  x.  The  index  is  4,  therefore 
there  must  be  five  places  for  whole  numbers.  The  decimal  part 
of  the  log.  is  a  little  greater  than  the  log.  of  4,  therefore  the 
number  sought  must  be  a  little  greater  than  40000.  It  is  not 
44000.  For  the  log.  of  11-|-  the  log.  of  4  will  give  a  greater 
decimal  than  the  decimal  .641519. 

»f  a  logarithm,  dy  the  differential  value  of  the  number,  and  y  the  number 
tself. 

The  last  operation  in  (Art.  15),        y=2940,        m=.434294. 
m.4342944 


=00()0148     rfjr=0  oom6.    dy=-^*l3  nearly. 
y        2940  m         .000148 

When  y=l,  then  dx—tndy,  and  dy=  —  ,  which  is  the  symbol  for  the  last 


*nle. 


42  SURVEYING. 

The  number  42,  or  42000,  which  is  probably  less  than  a?,  has 
three  factors,  6,  7,  and  1000. 

The  log.  of  6=log.2-}-log.3,         -        -      .778151 

log.  7,     -         -          .845098 

Log.  1000+  log.  42,  -  4.623249 

Log.  ar,         -         -      4.641519 

L°g'  42000  0.018270 

The  number  corresponding  to  this  remaining  log.  is  found  by 
the  last  rule. 

01  ft97 

Thus  =0.04205.         Log.  of  1.04205  is  ,018270. 

.43429 

Whence       — —  =1.04205,  or  z=43766.1,  the  approximate 
42000 

number. 

It  is  not  likely,  however,  that  this  is  the  true  number,  for  the 
log.  0.018270,  is  too  great  to  have  its  number  determined  by 
this  method,  and  if  greater  accuracy  is  required,  we  must  obtain 
more  exact  factors. 

We  will  therefore  resume  the  work  from 

log       x     =0.018270. 
3   42000 

and  we  know  that  the  number  corresponding  to  this  log.  must 
be  near  1 .042  ,  and  this  number,  or  a  number  very  near  it,  can 
be  produced  by  dividing  100  by  96.  Or,  in  other  words,  this 
log.  must  be  very  near  the  log.  of  VV  • 

But  the  log.  of  100  is  2.000000 

Log.  of  96  is  log.  of  8+log.  2+log.6,  1 .982271 


0.017729 

Making  use  of  this  factor,  we  shall  have 

Log.      96*  -=0.000541. 
3   4200000 

Now  we  have  a  log.  sufficiently  small  to  determine  its  number, 
as  follows  : 

I000541  =0.001246.  Log.  of  1.001246=0.000541. 
•434,69 


LOGARITHMS.  43 

Whence  _J5f—=  1.00 1246. 
4200000 

Or  J^L=  100. 1246.        Or    s=43804.516, 

7000 

the  true  number  sought. 

But  by  the  help  of  the  table  this  problem  would  have  been  a 
very  trifling  affair ;  but  if  a  person  can  work  it  without  the  table 
by  recollecting  a  few  simple  logarithms,  we  are  sure  that  such  a 
person  has  a  clear  comprehension  of  logarithms,  and  it  is  to 
make  this  test  that  we  require  any  one  to  work  without  the  table. 

The  following  examples  are  given  to  show  the  practical  value 
of  logarithms.  The  student  may  now  use  the  table. 

EXAMPLES. 

1.    What  is  the  cube  root  of  12326391?  Ans.  231. 

The  log.  of  12320000  is  7.090611     (See  table.) 

Correction  for  .6391  225 


Log.  of  12326391  3)7.090836 

Log.  of  231  2.363612 

2.  What  is  the  cube  root  of  592.704?  Ans.  8.4. 

3.  What  is  the  cube  root  of  997002999?  Ans.  999. 

4.  What  is  the  cube  root  of  40?  Ans.  3.41995-J-. 
The  utility  of  logarithms  is  more   strikingly  illustrated  by 

examples  in  powers  and  roots  higher  than  the  third,  like  the  fol- 
lowing : 

5.  What  is  the  5th  root  of  130691232?  Ans.  42. 
Log.  of  130691232  is                                   8.116246(5 

Log.  of  42  is  1.623249 

6.  What  is  the  6th  root  of  12230590464?  Ans.  48. 

7.  What  is  the  7th  root  of  10?  Ans.   1.38949. 

8.  The  10th  root  of  a  certain  number  is  .021,  what  is  that 
number?  Ans.  0.0000.0000.0000.00001 6684+. 

9.  The  10th  root  of  a  certain  number  is  21,  what  is  that 
number?  Ans.  16684000000000,  nearly. 


14  SURVEYING. 


CHAPTER    III. 

ELEMENTARY    PRINCIPLES    OF    PLANE 
TRIGONOMETRY. 

TRIGONOMETRY  in  its  literal  and  restricted  sense,  has  for  its  object, 
the  measure  of  triangles.  When  the  triangles  are  on  planes,  it  is 
plane  trigonometry,  and  when  the  triangles  are  on,  or  conceived  to 
be  portions  of  a  sphere,  it  is  spherical  trigonometry.  In  a  more 
enlarged  sense,  however,  this  science  is  the  application  of  the  prin- 
ciples of  geometry,  and  numerically  connects  one  part  of  a  magni- 
tude with  another,  or  numerically  compares  different  magnitudes. 

As  the  sides  and  angles  of  triangles  are  quantities  of  different 
kinds,  they  cannot  be  compared  with  each  other ;  but  the  relation 
may  be  discovered  by  means  of  other  complete  triangles,  to  which 
the  triangle  under  investigation  can  be  compared. 

Such  other  triangles  are  numerically  expressed  in  Table  II,  and 
all  of  them  are  conceived  to  have  one  common  point,  the  center  of 
a  circle,  and  as  all  possible  angles  can  be  formed  by  two  straight 
lines  drawn  from  the  center  of  a  circle,  no  angle  of  a  triangle  can 
exist  whose  measure  cannot  be  found  in  the  table  of  trigonometrical 
lines. 

The  measure  of  an  angle  is  the  arc  of  a  circle,  intercepted  be- 
tween the  two  lines  which  form  the  angle — the  center  of  the  arc 
always  being  at  the  point  where  the  two  lines  meet. 

The  arc  is  measured  by  degrees,  minutes,  and  seconds,  there  being 
360  degrees  to  the  whole  circle,  60  minutes  in  one  degree,  and  60 
seconds  in  one  minute.  Degrees,  minutes,  and  seconds,  are  desig- 
nated by  °,  ',  ".  Thus  27°  14'  21",  is  read  27  degrees,  14  min- 
utes, and  21  seconds. 

All  circles  contain  the  same  number  of  degrees,  but  the  greater 
the  radii  the  greater  is  the  absolute  length  of  a  degree  ;  the  cir- 
cumference of  a  carriage  wheel,  the  circumference  of  the  earth,  or 
the  still  greater  and  indefinite  circumference  of  the  heavens,  have 
the  same  number  of  degrees  ;  yet  the  same  number  of  degrees  in 
each  and  every  circle  is  precisely  the  same  angle  in  amount  or 
measure. 


PLANE    TRIGONOMETRY.  45 

As  triangles  do  not  contain  circles,  we  can  not  measure  triangles 
by  circular  arcs  ;  we  must  measure  them  by  other  triangles,  that  is, 
by  straight  lines,  drawn  in  and  about  a  circle. 

Such  straight  lines  are  called  trigonometrical  lines,  and  take  par- 
ticular names,  as  described  by  the  following 

DEFINITIONS. 

1.  The  sine  of  an  angle,  or  an  arc,  is  a  line  drawn  from  one  end 
of  an  arc,  perpendicular  to  a  diameter  drawn  through  the  other  end. 
Thus,  BF  is  the  sine  of  the  arc  AB,  and  also  of  the  arc  BD E.    BK 
is  the  sine  of  the  arc  BD,  it  is  also  the  cosine  of  the  arc  AB,  and 
BF,  is  the  cosine  of  the  arc  BD. 

N.  B.  The  complement  of  an  arc  is  what  it 
wants  of  90° ;  the  supplement  of  an  arc  is 
what  it  what  it  wants  of  180°. 

2.  The  cosine  of  an  arc  is  the  perpendicu- 
lar distance  from  the  center  of  the  circle  to 
the  sine  of  the  arc,  or  it  is  the  same  in  mag- 
nitude as  the  sine  of  the  complement  of  the 

arc.     Thus,  CF,  is  the  cosine  of  the  arc  AB;  but  QF=KB,  the 
sine  of  BD. 

3.  The  tangent  of  an  arc  is  a  line  touching  the  circle  in  one 
extremity  of  the  arc,  continued  from  thence,  to  meet  a  line  drawn 
through  the  center  and  the  other  extremity. 

Thus,  AH  is  the  tangent  to  the  arc  AB,  and  DL  is  the  tangent 
of  the  arc  DB,  or  the  cotangent  of  the  arc  AB. 

N.  B.  The  co,  is  but  a  contraction  of  the  word  complement. 

4.  The  secant  of  an  arc,  is  a  line  drawn  from  the  center  of  the 
circle  to  the  extremity  of  its  tangent.     Thus,  CH  is  the  secant  of 
the  arc  AB,  or  of  its  supplement  BDK 

5.  The  cosecant  of   an  arc,  is  the  secant  of  the  complement. 
Thus,  CL,  the  secant  of  BD,  is  the  cosecant  of  AB. 

6.  The  versed  sine  of  an  arc  is  the  difference  between  the  cosine 
and  the  radius  ;  that  is,  AF  is  the  versed  sine  of  the  arc  AB,  and 
DK  is  the  versed  sine  of  the  arc  BD. 

For  the  sake  of  brevity  these  technical  terms  are  contracted  thus : 
for  sine  AB,  we  write  sin.AB,  for  cosine  AB,  we  write  cos.AB, 
for  tangent  AB,  we  write  tan.AB,  <fcc. 


46  SURVEYING. 

From  the  preceding  definitions  we  deduce  the  following  obvious 
consequences : 

1st,  That  when  the  arc  AB,  becomes  so  small  as  to  call  it 
nothing,  its  sine  tangent  and  versed  sine  are  also  nothing,  and  its 
secant  and  cosine  are  each  equal  to  radius. 

2d,  The  sine  and  versed  sine  of  a  quadrant  are  each  equal  to  the 
radius  ;  its  cosine  is  zero,  and  its  secant  and  tangent  are  infinite. 

3d,  The  chord  of  an  arc  is  twice  the  sine  of  half  the  arc.  Thus, 
the  chord  BQ,  is  double  of  the  sine  BF. 

4th,  The  sine  and  cosine  of  any  arc  form  the  two  sides  of  a 
right  angled  triangle,  which  has  a  radius  for  its  hypotenuse.  Thus, 
GFy  and  FB,  are  the  two  sides  of  the  right  angled  triangle  CFB. 

Also,  the  radius  and  the  tangent  always  form  the  two  sides  of  a 
right  angled  triangle  which  has  the  secant  of  the  arc  for  its  hypo- 
tenuse. This  we  observe  from  the  right  angled  triangle  GAH. 

To  express  these  relations  analytically,  we  write 

^2  (1) 

sec.2  (2) 

From  the  two  equiangular  triangles  CFB,  CAR,  we  have 
CF:FB=CA'.AH 

That  is,  cos.  :  sin=jR :  tan.  tan.= •'     (3) 

cos.       v  ' 

Also,  .        CF\CB=CA:CH 

That  is,        .  cos  :  R=R :  sec.        cos.  sec. =^2       (4) 

The  two  equiangular  triangles  CAff,  CDL.  give 
CA:Aff=DL:DG 

That  is,        .  R :  tan.=cot :  R          tan.  cot^JP    (5) 

Also,  .         CF :  F£=DL :  D  C 

That  is,        .         cos. :  sin.=cot :  R      cos.  J2=sin.  cot.     (6) 

By  observing  (4)  and  (5),  we  find  that 

cos.  sec.=tan.  cot.  (1) 

Or,  .         cos.  :tan.=cot.  :sec. 

The  ratios  between  the  various  trigonometrical  lines  are  always  the 
same  for  the  same  arc,  whatever  be  the  length  of  the  radius  ;  ana 
therefore,  we  may  assume  radius  of  any  length  to  suit  our  conven- 
ience ;  and  the  preceding  equations  will  be  more  concise,  and  more 


PLANE    TRIGONOMETRY.  47 

readily  applied,  by  making  radius  equal  unity.  This  supposition 
being  made,  the  preceding  become 

sin.24-cos.2=l  (1) 

l+tan.2=sec.a  (2) 

tan.= — *     (3)  cos.= —  (4) 

cos.  sec. 

tan.=— -     (5)  cos.=sin.  cot.     (6) 

The  center  of  the  circle  is  considered  the  absolute  zero  point,  and 
the  different  directions  from  this  point  are  designated  by  the  different 
signs  -f-  and  — .  On  the  right  of  (7,  toward  A,  is  commonly 
marked  plus  (+),  then  the  other  direction,  toward  E,  is  necessarily 
minus  ( — ).  Above  .^LZ^is  called  (4),  below  that  line  ( — ). 

If  we  conceive  an  arc  to  commence  at  A,  and  increase  contin- 
uously around  the  whole  circle  in  the  direction  of  ABD,  then  the 
following  table  will  show  the  mutations  of  the  signs. 

sin.         cos.        ten.         cot.         sec.      cosec.       vers. 
1st  quadrant,    -f         +         +         +         +         4-         4- 
2d         "  +  4-4- 

3d  +         +        __        _        4. 

4th       "  —        4-  4-  4- 

PROPOSITION   1. 

The  chord  of  60°  and  the  tangent  45°  are  each  equal  to  radius; 
the  sine  of  30°  the  versed  sine  of  60°  and  the  cosine  of  60°  are  each 
equal  to  half  the  radius. 

(The  first  truth  is  proved  in  problem  15,  book  1). 

On  C=,  as  radius,  describe  a  quadrant ;  take  AD=45°,  AB 
=60°,  and  AE=90°,  then  B£=3Q°. 

Join  AB,  CB,  and  draw  Bn,  perpendicular  to  CA.  Draw  Bm, 
parallel  to  AC.  Make  the  angle  (L4ZT=90°,  and  draw  CDH. 

In  the  A  ABC,  the  angle  ACB—QO0 
by  hypothesis ;  therefore,  the  sum  of  the 
other  two  angles  is  ( 1 80—60) = 1 20° .  But 
CB=  CA,  hence  the  angle  CBA=  the  angle 
CAB,  (th.  1 5  b.  1 ) ,  and  as  the  sum  of  the  two 
is  120°,  each  one  must  be  60° ;  therefore, 
each  of  the  angles  of  triangle  ABC,  is  60° 


48  SURVEYING. 

and  the  sides  opposite  to  equal  angles  are  equal ;  that  is,  AB,  the 
chord  of  60°,  is  equal  to  CA,  the  radius. 

In  the  A  CAff,  the  angle  CAff  is  a  right  angle ;  and  by  hypoth- 
esis, A  Clf,  is  half  a  right  angle ;  therefore,  AffC,  is  also  half  a  right 
angle ;  consequently,  Aff=AC,  the  tangent  of  45°=  the  radius. 

By  th.  15,  book  1,  cor.  Cn=nA;  that  is,  the  cosine  and  versed 
sine  of  60°  are  each  equal  to  the  half  of  the  radius.  As  Bn  and 
EC  are  perpendicular  to  A  C,  they  are  parallel,  and  Bm  is  made 
parallel  to  On;  therefore,  Bm=Cn,  or  the  sine  30°,  is  the  half  of 
radius. 

PROPOSITION     2. 

Given  the  sine  and  cosine  of  two  arcs  to  find  the  sine  and  cosine  of 
the  sum,  and  difference  of  the  same  arcs  expressed  by  the  sines  and  co- 
sines of  the  separate  arcs. 

Let  G  be  the  center  of  the  circle,  CD,  the 
greater  arc  which  we  shall  designate  by  a, 
and  DF,  a  less  arc,  that  we  designate  by  b. 

Then  by  the  definitions  of  sines  and  co- 
sines, Z>0=sin.a;    G0=cos.a;  FI=sm.b; 
@Z=cos.b.     We  are  to  find  FM,  which  is 
#J/"=cos.(a-f  b); 
#P=cos.(a— • b). 

Because  IN  is  parallel  to  D  0,  the  two  AS  GD  0,  G-IN,  are 
equiangular  and  similar.  Also,  the  A  FBI,  is  similar  to  GIN; 
for  the  angle  FIG,  is  a  right  angle  ;  so  is  HIN;  and,  from  these 
two  equals  take  away  the  common  angle  IIIL,  leaving  the  angle 
FIH=  GIN.  The  angles  at  H  and  N,  are  right  angles  ;  therefore, 
the  A  FHI,  is  equiangular,  and  similar  to  the  A  GIN,  and,  of 
course,  to  the  A  GD  0;  and  the  side  HI,  is  homologous  to  IN, 
and  D  0. 

Again,  as  FI=IE,  and  IK,  parallel  to  FM, 
FH=IK,  and  HI=KE. 

By  similar  triangles  we  have 

QD:DO=GI:IN. 

That  is,  £:sm.a=coa.b:W,   or  Itf=*mM pC°8  - 

jK 

Also,  GD-.GO=FI'.FB 


PLANE    TRIGONOMETRY.  49 

That  is,  R :  cos.a=sin.5 :  Fff,  or  FH=  — ^ — - 

Also,  GD:GO=GI:GN 

,    cos.a  cos.5 
That  is,  R :  cos.a=cos.5 :  GN,  or  GN— — -= 

Also,  GD:DO=FI:IH 

rTT    sin.a  sin.5 
That  is,  R  :  sm.a=sin.5 :  IB,    or  Iff= -^ 

By  adding  the  first  and  second  of  these  equations,  we  have 

,    ,           sin.a  cos.5+cos.a  sin.5 
That  is,        .      sin.  (a-f  5)=- ~ 

By  subtracting  the  second  from  the  first,  we  have 

sin.a  cos.5 — cos.a  sin.5 
sin. 


By  subtracting  the  fourth  from  the  third,  we  have 

GN—IH=  GM=cos.(a-\-b)  for  the  first  member. 

TX     cos.a  cos.b  —  sin.a  sin.6 
Hence,          .      cos.(a-f-o)=  -  ^  - 

By  adding  the  third  and  fourth,  we  have 

=  GP=cos.(a—b) 


_x   cos.a  cos. 
Hence,          .      cos.  (a  —  b)= 


R 

Collecting  these  four  expressions,  and  considering  the  radius 
unity,  we  have 

sin.(a-{-5^=sin.a  cos.J+cos.a  sin.5  £7) 

sinYa— 5)=sm.a  cos.5— -cos.a  sm.6  r8j 

cos.(a+6J=cos.a  cos.6 — sin.a  sin.5  (9) 
cos. (a---5)=cos.a  cos. 5-|-sin.a  sin.5       (10) 

Formula  (A),  accomplishes  the  objects  of  the  proposition,  and 
from  these  equations  many  useful  and  important  deductions  can  be 
made.  The  following,  are  the  most  essential : 

By  adding  (7)  to  (8),  we  have  (11);  subtracting  (8)  from  (7), 
gives  (12).  Also,  (9)+(10)  gives  (13);  (9)  taken  from  (10) 
gives  (14). 

cos  5  (11 


(B\    i    sin.ra4-Jl — sin.(a — 5)=2cos.a  sin.  5  H2 

*    '     i    cos.?a-(-5)+cos.(a — 5)=2cos.a  cos.5  (13 

cos. (a— 5) — cos.(a+5)=2sin.  a  sin.5  (14 


50 


SURVEYING. 


If  we  put  a-\-b=A,  and  a — b=J3,  then  (11)  becomes  (15), 
(12)  becomes  (16),  13  becomes  (17),  and  (14)  becomes  (18). 


cos. 


(15) 


sm. 


m.£=2cos.  (         -  )  sin.   (       p    )      (16) 
(  ^=?  )      (17) 


cos.-4-|-cos.J5=2cos 


cos. 


sin.  (18) 


If  we  divide  (15)  by   (16),  (observing  that  -— =tan.   and 

COS* 

— =cot.=- —  as  we  learn  by  equations  (6)  and  (5)  trigonome- 
sin.  tan* 


),  we  shall  have 
sin.-4-f-sin.5 


Whence, 


(19) 


:  sin.^— sin..B=tan.  ^  —  —  -  j  :  tan.  ^  —  — 

or  in  words.  The  sum  of  the  sines  of  any  two  arcs  is  to  the  differ- 
ence of  the  same  sines,  as  the  tangent  of  the  half  sum  of  the  same  arcs 
is  to  the  tangent  of  half  their  difference. 

By  operating  in  the  same  way  with  the  different  equations  in  for- 
mula (0),  we  find, 

(20) 
(21) 
(22) 
(23) 


cos..4-|-cos.-B  \      2      / 

..5  /  A  —  B  \ 

,=cot.  (  —  —  -  ) 
s.^4  \      2      / 

in..g  _         /  A  —  B  \ 
cos.^l-l-cos7S~  an*  I  "T"  j 


cos. 


4=cot.(^±f) 
COS..D — cos.  .4  \      2      / 

cos.A-\-cos.JB 
cos.2? — cos.Jf 


. lA+B\ 
COt-(—  ) 


tan. 


(24) 


PLANE    TRIGONOMETRY  51 

These  equations  are  all  true,  whatever  be  the  value  of  the  arcs 
designated  by  A  and  B;  we  may  therefore,  assign  any  possible 
value  to  either  of  them,  and  if  in  equations  (20),  (21)  and  (24), 
we  make  JS=  0,  we  shall  have, 

sin.  A  A         I 


1 +oos.^l  2      cot^A 

ALA  .A         1 


If  we  now  turn  back  to  formula  (A),  and  divide  equation  (7)  by 
),  and  (8)  b 
we  shall  have, 


(9),  and  (8)  by  (10),  observing  at  the  same  time,  that  —  -=tan. 

cos* 


sin  a  cos.54-  cos.a  sin.6 

tan.(a-H)  =  --  —--.  -  -  -= 

cos  .a  cos.6  —  sm.a  sin.fi 

_  x     sin.a  cos.5  —  cos.a  sin.6 

tan.(a—  ^)=  -  =-=--:  -  ^~ 

cos.a  cos.o+sm.a  sm.b 

By  dividing  the  numerators  and  denominators  of  the  second 
members  of  these  equations  by  (cos.a  cos.fl),  we  find, 

sin.a  cos.5     cos.a  sin.6 

cos.a  cos.6  cos.a  cos.6  tan.a-|-tan.6 
-  -  --  :  __  r  _  =  _  -  _ 
cos.a  cos.6  sin.a  sin.6  1  —  tan.atan.6 

cos.a  cos.£  cos.ocos.6 

sin.a  cos.5  cos.a  sin.£ 

.  _  .  .  _cos.a  cos.6  cos.g  cos.^_  tan.a  —  tan.6 

*         ^~cos.q  cos.b  sin.a  sin.6'~'l-|-tan.a  tan.6      '     ' 

cos.a  cos.6  cos.a  cos.5 

If  in  equation  (11),  formula  (5),  we  make  a=5,  we  shall  hare, 

sin.2a=2sin.a  cos.a         (30) 
Making  the  same  hypothesis  in  equation  (13),  gives, 

cos.2a-f  I=2cos2.a  (31) 

The  Same  hypothesis  reduces  equation  (14),  to 

1  —  cos.2«=2sin2.a  (32) 

The  same  hypothesis  reduces  equation  (28),  to 

{33) 

v     ' 


52  SURVEYING. 

The  secants  and  cosecants  of  arcs  are  not  given  in  our  table,  because 
they  are  very  little  used  in  practice  ;  and  if  any  particular  secant  is 
required,  it  can  be  determined  by  subtracting  the  cosine  from  20  ;  and 
the  cosecant  can  be  found  by  subtracting  the  sine  from  20. 


PROPOSITION    3. 

In  any  right  angled  plane  triangle,  w  may  have  the  following 
proportions  : 

1st.  As  the  hypotenuse  is  to  either  side,  so  is  the  radius  to  the  sine 
of  the  angle  opposite  to  that  side. 

2d.  As  one  side  is  to  the  other  side,  so  is  the  radius  to  the  tangent 
of  the  angle  adjacent  to  the  first-mentioned  side. 

3d.  As  one  side  is  to  the  hypotenuse,  so  is  radius  to  the  secant  of 
the  angle  adjacent  to  that  side. 

Let  CAB  represent  any  right 
angled  triangle,  right  angled  at  A. 
AB  and  AG  are  called  the  sides 
of  the  A,  and  CB  is  called  the 
hypotenuse. 

(Here,  and  in  all  cases  hereafter,  we  shall  represent  the  angles  of  a  triangle 
by  the  large  letters  A,  B,  C,  and  the  sides  opposite  to  them,  by  the  small  letters 
a,  b,  c.) 

From  either  acute  angle,  as  C,  take  any  distance,  as  CD,  greater 
or  less  than  CB,  and  describe  the  arc  JDK  This  arc  measures 
the  angle  C.  From  D,  draw  DF  parallel  to  BA;  and  from.  J?, 
draw  EG,  also  parallel  to  BA  or  DF. 

By  the  definitions  of  sines,  tangents,  and  secants,  DF  is  the  sine 
of  the  angle  C  ;  EG-  is  the  tangent,  CO  the  secant,  and  CF  the 
cosine. 

Now,  by  proportional  triangles  we  have, 


CB  :  jBA=CD  :  DF  or,  a 
CA  :  AB=  CE  :  EG  or,  b 
CA  :  CB=CE  :  CG  or,  b 


=R  :  siu.C 


tan.  tf     Q.  E.  D. 


a=R  :  sec.CJ 


Scholium.  If  the  hypotenuse  of  a  triangle  is  made  radius,  one 
side  is  the  sine  of  the  angle  opposite  to  it,  and  the  other  side  is  the 
cosine  of  the  same  angle.  This  is  obvious  from  the  triangle  CDF. 


PLANE    TRIGONOMETRY.  53 

PROPOSITION    4. 

In  any  triangle,  the  sines  of  the  angles  are  to  one  another  as  the 
tides  opposite  to  them. 

Let  AE  C  be  any  tri- 
angle. From  the  points 
A  and  JB,  as  centers,  with 
any  radius,  describe  the 
arcs  measuring  these  an- 
gles, and  draw  pa,  CD, 
and  mn,  perpendicular  to  AB. 

Then,         .         .  pa=sm.A,  mn=Bm.JB 

By  the  similar  AS,  Apa   and  A  CD,  we  have, 

R  :  sin.A=b  :  CD;  or,  JR(CD)=b  sin. A    (1) 

By  the  similar  AS  Bmn  and  BCD,  we  have, 

R  :  sm.B=a  :  CD;  or,  fi(CD)=asm.£    (2) 
By  equating  the  second  members  of  equations  (1)  and  (2). 
b  sm.A=a  sin.J?. 

Hence,      .  sm.A  :  sm.B=a  :  b  \   O  E  D 

Or,   .  a  :  b=sm  A  :  sin.  B} 

Scholium  1.  When  either  angle  is  90°,  its  sine  is  radius. 

Scholium  2.  When  CB  is  less  than  A  C,  and  the  angle  B,  acute, 
the  triangle  is  represented  by  ACB.  When  the  angle  B  becomes 
B',  it  is  obtuse,  and  the  triangle  is  A  CB';  but  the  proportion  is 
equally  true  with  either  triangle ;  for  the  angle  C£'D=  CBA, 
and  the  sine  of  CB'D  is  the  same  as  the  sine  of  AB'C.  In  prac- 
tice we  can  determine  which  of  these  triangles  is  proposed  by  the 
side  AB,  being  greater  or  less  than  A  C;  or,  by  the  angle  at  the 
vertex  C,  being  large  as  A  CB,  or  small  as  A  GB'. 

In  the  solitary  case  in  which  A  C,  CB,  and  the  angle  A,  are  given, 
and  CB  less  than  A  C,  we  can  determine  both  of  the  As  A  CB 
and  A  CB';  and  then  we  surely  have  the  right  one. 

PROPOSITION    5. 

If  from  any  angle  of  a  triangle,  a  perpendicular  be  let  fall  on  the 
opposite  side,  or  base,  the  tangents  of  the  segments  of  the  angle  are  to 
one  another  as  the  segments  of  the  base. 


54  SURVEYING 

Let  ABC  be  the  triangle.  Let  fall  the 
perpendicular  CD,  on  the  side  AB. 

Take  any  radius,  as  Cn,  and  describe 
the  arc  which  measures  the  angle  C. 
From  n,  draw  qnp  parallel  to  AB.  Then 
it  is  obvious  that  np  is  the  tangent  of  the 
angle  D  CB,  and  nq  is  the  tangent  of  the  angle  A  CD. 

Now,  by  reason  of  the  parallels  AB  and  qp,  we  have, 
qn  :  np=AD  :  DB 

That  is,    tea.  A  CD  :  t&n.DC£=AD  :  DB        Q.  E.  D. 

PROPOSITION     6. 

If  a  perpendicular  be  let  fall  from  any  angle  of  a  triangle  to  its  op- 
posite side  or  base,  this  base  is  to  the  sum  of  the  other  two  sides,  as  the 
difference  of  the  sides  is  to  the  difference  of  the  segments  of  the  base. 
(See  figure  to  proposition  5.) 

Let  AB  be  the  base,  and  from  (7,  as  a  center,  with  the  shorter 
side  as  radius,  describe  the  circle,  cutting  AB  in  G,  AC  in  F,  and 
produce  A  C  to  E. 

It  is  obvious  that  AE  is  the  sum  of  the  sides  A  C  and  CB,  and 
AF  is  their  difference. 

Also,  AD  is  one  segment  of  the  base  made  by  the  perpendicular, 
and  BD—DG  is  the  other;  therefore,  the  difference  of  the  seg- 
ments is  A  G. 

As  A  is  a  point  without  a  circle,  by  theorem  1  8,  book  3,  we  have, 


Hence,        .        .    AB  :  AE=AF  :  AG        Q.  E.  D. 

PROPOSITION    7. 

The  sum  of  any  two  sides  of  a  triangle,  is  to  their  difference,  as 
the  tangent  of  the  half  sum  of  the  angles  opposite  to  these  sides,  to 
the  tangent  of  half  their  difference. 

Let  ABC  be  any  plane   triangle.     Then, 
by  proposition  4,  trigonometry,  we  have, 

CB  :  AC=siu.  A  :  sm.B 
Hence, 
CB+AC  :  CB—AC=sm.A+sm.£  :  sm.A—  sin.B  (th.  9  b.  2) 


But,  tan. 


PLANE    TRIGONOMETRY. 

.A— B 

:  tan 


55 


(eq.(l),trig.) 

Comparing  the  two  latter  proportions  (th.  6,  b.  2),  we  have, 

):tan.(±I-)  Q.  KD. 


CJB+A C  :  CJB—AC=  tan.  ( 


PROPOSITION    8. 

Given  the  three  sides  of  any  plane  triangle,  to  find  some  relation 
which  they  must  bear  to  the  sines  and  cosines  of  the  respective  angles. 

Let  ABC  be  the 

triangle,  and  let  the 
perpendicular  fall 
either  upon,  or 
without  the  base, 
as  shown  in  the 
figures ;  and  by 
recurring  to  theorem  38,  book  1,  we  shall  find 


2a 

Now,  by  proposition  3,  trigonometry,  we  have, 
R\  cos.  (7=5:  CD 
b  cos.  C 


Therefore, 


CD=- 


R 


(2) 


Equating  these  two  values  of  CD,  and  reducing,  we  have, 


In  this  expression  we  observe  that  the  part  of  the  numerator 
which  has  the  minus  sign,  is  the  side  opposite  to  the  angle  ;  and 
that  the  denominator  is  twice  the  rectangle  of  the  sides  adjacent 
to  the  angle.  From  these  observations  we  at  once  draw  the  fol- 
lowing expressions  for  the  cosine  A,  and  cosine  B. 


Thus, 


(n) 


COS.-B: 


(P) 


SURVEYING. 


As  these  expressions  are  not  convenient  for  logarithmic  compu- 
tation, we  modify  them  as  follows  : 

If  we  put  2a=-4,  in  equation  (31),  we  have, 


In  the  preceding  expression  (n),  if  we  consider  radius,  unity, 
and  add  1  to  both  members,  we  shall  have, 


Therefore, 


Considering  (6-f-c  )  as  one  quantity,  and  observing  that  we  have 
the  difference  of  two  squares,  therefore 

-  a)(H-  c—  a);  but  (b+  c—a)=b-{-c+a—  2a 


Hence.        .     2  cos.' 
Or>     .        .        eos 


w 

By  putting =*,  and  extracting  square   root,  the  final 

result  for  radius  unity,  is 


For  any  other  radius  we  must  write, 
cos.k4= 


T>      •    f  1  *          l&s(S—b) 

By  inference,          cos.+E=\l * - 

N        ac 

iW* 
Also,         .        .     cos-Atf 


ab 

In  every  triangle,  the  sum  of  the  three  angles  must  equal  180°;  and 
if  one  of  the  angles  is  small,  the  other  two  must  be  comparatively 
large;  if  two  of  them  are  small,  the  third  one  must  be  large.  The 
greater  angle  is  always  opposite  the  greater  side ;  hence,  by  merely 
inspecting  the  given  sides,  any  person  can  decide  at  once  which  is  the 
greater  angle ;  and  of  the  three  preceding  equations,  that  one  should 
be  taken  which  applies  to  the  greater  angle,  whether  that  be  the  par- 
ticular angle  required  or  not ;  because  the  equations  bring  out  the 


PLANE    TRIGONOMETRY.  57 

cosines  to  the  angles  ;  and  the  cosines,  to  very  small  arcs  vary  so  slowly, 
that  it  maybe  impossible  to  decide,  with  sufficient  numerical  accuracy 
to  what  particular  arc  the  cosine  belongs.  For  instance,  the  cosine 
9.999999,  carried  to  the  table,  applies  to  several  arcs  ;  and,  of  course, 
we  should  not  know  which  one  to  take  ;  but  this  difficulty  does  not  exist 
when  the  angle  is  large  ;  therefore,  compute  the  largest  angle  first, 
and  then  compute  the  other  angles  by  proposition  4. 

But  we  can  deduce  an  expression  for  the  sine  of  any  of  the  angles, 
as  well  as  the  cosine.     It  is  done  as  follows  : 

EQUATIONS    FOR    THE    SINES    OF    THE    ANGLES. 
Resuming  equation  (m),  and  considering  radius,  unity,  we  hare, 

a'-H2—  c2 
cos.  C=  —         •  • 
2ab 

Subtracting  each  member  of  this  equation  from  1,  gives 


. 

Making  2a=(7,  in  equation  (32),  then  a=-J<7, 

And        .        .   1  —  cos.  (7=  2  sin.2£(7 

Equating  the  right  hand  members  of  (1)  and  (2), 


Or,    .        .        .Sin.»|<7 


Put   .          '• — - — =8,    as  before ;  then, 


)(s—b) 
/v        '- 


By  taking  equation  (p),  and  operating  in  the  same  manner*  we 
have  ,    sin.-kS 


From  (»)  .        .  8iB.^= 


58  SURVEYING. 

The  preceding   results    are    for   radius   unity;    for  anj  other 
radius,  we  must  multiply  by  the  number  of  units  in  such  radhr 
For  the  radius  of  the  tables,  we  write  R;  and  if  we  put  it  under 
the  radical  sign,  we  must  write  M2;  hence,  for  the  sines  corres- 
ponding with  our  logarithmic  table,  we  must  write  the  equations 


thus, 


—  b)(s—  c) 
—  '^  -  '- 


A  large  angle  should  not  be  determined  by  these  equations,  for 
the  same  reason  that  a  small  angle  should  not  be  determined  from 
an  equation  expressing  the  cosine. 

In  practice,  the  equations  for  cosine  are  more  generally  used, 
because  more  easily  applied. 

In  the  preceding  pages  we  have  gone  over  the  whole  ground  of 
theoretical  plane  trigonometry,  although  several  particulars  might 
have  been  enlarged  upon,  and  more  equations  in  relation  to  the 
combinations  of  the  trigonometrical  lines,  might  have  been  given  ; 
but  enough  has  been  given  to  solve  every  possible  case  that  can  arise 
in  the  practical  application  of  the  science. 

By  the  application  of  equations  (1),  (31),  and  (32),  the  table 
of  natural  sines  and  cosines  has  been  computed. 

The  operation  is  as  follows.     The  sine  of  30°  is  half  radius  ; 
making  the  radius  unity,  equation  (1)  gives 
i+cos.2  30°=  1  :  whence  cos.2  30°=£  or  cos.  30°=^  J$ 

From  (32)  we  have,    sm.q==A/  1~~cos-  2q 

2 
Making  2a=30°,  then  sin.  150=(^—^JS/§)*=0.25881904 

From  (31)  we  have,    cos.g=/y//1+cos.  2g 

Making  2a=30°  as  before,  cos.a=(£-|-  J«/3  )*=0.96592582 
Having  sine  and  cosine  of  15°  the  second  application  of  these 

equations  will  give  the  she  and  cosine  of  the  half  of  15°,  and  so 

on  through  as  many  bisections  as  we  please. 


PLANE    TRIGONOMETRY.  59 

Being  desirous  of  giving  a  full  exposition  of  the  formation  of 
table  II,  we  give  the  following  geometrical  demonstration  of  equa- 
tion 30,  by  the  help  of  the  figure  in  the  margin. 

Let  the  arc  A£=2a 
Then  DG=sm.  2a,     CG=cos.  2a, 
DI  =sin.  a,     AD—%  sin.  a, 
GI  *=cos.  a,     DB  =2J9  0=2  cos.  a. 
The  angle  DBA  being  at  the  circum- 
ference, is  measured  by  half  the  arc  AD, 
or  by  a. 

Now,  by  applying  proposition  4  to  the  triangle  DBG,  we  have 
sin.  DBG  :  DG=sm.  90°  :  BD. 

The  sin.  DBG=sin.  a,  and  sin.  90°=  1,  the  radius  being  unity; 
therefore,  the  preceding  proportion  becomes, 

ski.  a  :  sin.  2a=l  :  2  cos.  a. 
Whence  2  cos.  a  sin.  a=sin.  2a.        (Same  as  eq.  30.) 

PEOBLEM. 

Given  the  sine  and  cosine  of  an  arc,  to  find  the  sine  and  cosine  of 
one  half  that  arc. 

Designate  the  given  arc  by  2a,  the  radius  by  unity,  and  whatever 
be  the  value  of  a,  equation  (1)  gives 

cos.2a-{-sm.2a=l  (m) 

It  is  proved  in  proposition  1,  that  the  sine  of  30°  is  half  the  radius: 
therefore,  let  2a=30°,  then  sin.  2a=0.5  :  and  equation  30,  just 
demonstrated,  gives 

2  cos.  a  sin.  a=0.  5.  (n) 

Add  (m)  and  (n),  and  extract  the  square  root  of  both  members. 

Then  cos.  a  -f  sin.  a =1.22474486  (o) 

Subtracting  (n)  from  (m),  and  extracting  square  root,  gives 

cos.  a — sin.  0=0.70710678  (p) 

By  subtracting,  and  adding  (p)  and  (o),  and  dividing  by  2,  we 
find 

sin.  o=sin.  15°=0.25881904 
cos.  a=cos.l5°=0.96592582 


60  SURVEYING. 

Nowlet2a=15°.     Then 

cos.aa-j-sin.2a=l. 

and  2  cos.  a  sin.  a=0.2588 1904 

Operating  as  before,  we  find 

,,~, sin.  a=sin.  7°  30'=0.  130526 192 

;!  eosl  <i=cos.  7°  30'=0.99 14447879 
5  Again,  put  2a=7°  30'  then  as  before, 

cos.2a-{-sin.2a=l 
v'«fcos.  a  sin.  a=0.1305261921 

These  equations  give 

sin.  <z=sin.  3°  45'=0.0654031291 
cos.  a=cos.  3°  45'=:0.9978589222 

Thus  we  can  bisect  the  arc  as  many  times  as  we  please.  After 
five  more  bisections,  we  have 

sin.  a=sin.  7'  1"  52 £'" =0.0020453077 
cos.  a=cos.  7'  1"  52f"=0.99999799 

As  the  sines  of  all  arcs  under  10',  may  be  considered  as  coin- 
ciding with  the  arc,  and  varying  with  it,  we  can  now  find  the  she 
of  1'  by  proportion. 

Thus,         7'  1"  52  J'"  :  1'      :  :  0.0020453077  :  sin.  1 
Or,  25312.5'"  :  3600  : : 

Or,  10125  :  1440  :  :  0.0020453077  :  sin.  1' 

Whence  sin.  l'=0.0002908882 

sin.  2'=0.00058 17764 
sin.  3'=0.0008726646 
In  formula  (.5)  equation  (11),  we  find 

sin.  (a-H)  +sm-  (a — b) =2  sin.  a  cos.  b 
Now,  if  a=3'  and  5=1' 

sin.  4'-}-sin.  2'=2  sin.  3'  cos.  1' 

We  have  already  sin.  2'  and  sin.  3',  and  cos.  1'  does  not  sensibly 
differ  from  unity,  therefore 

sin.  4'=2  sin.  3' — sin,  2'=0.001 1635528 
sin.  5'=2  sin.  4'  cos.  1 — sin.  3'  <fec.  <fec.  to  15' 
When  the  sine  of  any  arc  is  known,  its  cosine  can  be  found  by 
the  following  formula,  which  is,  in  substance,  equation  (1)  trigo- 
nometry cos.  a=  J(  1  -f-sin.  a)  ( 1 — sin.  a) 


PLANE    TRIGONOMETRY  61 

In  formula  (A)  equation  (7)  we  find  that 

sin.  (a-|-5)=sin.  a  cos.  5-{-cos.  a  sin.  b 
Now,  if  we  make  a=30°  and  5=4'  Then 

sin.  a=0.5        cos.  0=^^/3=0.8660254 
sin.  5=0.001 16355  cos.  5=0.999999323 
Whence 

sin.  (30°  4')=(0.5)  (0,999999323)+(0.8660254)  (0.00116355) 
=0.499999661  +  0.0010007620 

=0.501007281 
Equation  (8)  gives 

sin.  (29°  56')=0.498992041 

When  the  sine  and  cosine  of  any  arc  are  both  known,  the  sine 
and  cosine  of  the  half  or  double  of  the  arc  can  be  determined  by 
equation  30;  —  and  thus,  from  equations  (30),  (7),  (8),  (11),  and 
(1),  the  sines  and  cosines  of  all  arcs  can  be  determined. 

But  these  sines  and  cosines  are  expressed  in  natural  numbers,  to 
radius  unity,  hence  they  are  called  natural  sines  and  natural  cosines, 
and  they  are  all  decimals,  except  the  sine  of  90°  and  the  cosine  of 
0°,  each  of  which  is  unity. 

To  form  table  II,  we  require  logarithmic  sines,  and  cosines,  which 
are  found  by  taking  the  logarithms  of  the  natural  shies  and  cosines, 
and  increasing  the  indices  by  10,  to  correspond  to  the  radius  of 
10000000000.  The  radius  of  this  table  might  have  been  greater  or 
less,  but  custom  has  settled  on  this  value. 

To  find  the  logarithmic  sine  of  1',  we  proceed  thus, 

Nat.  sin.  1/=0.0002908882          log.                       —4.  463726 
To  which  add  10. 

The  log.  sine  of  1',  therefore  is  6.  463  726 

Nat.  sin.  3'=0.0008726646          log.  —4.  950  847 

Add  10 

Log.  sin.  3'  therefore  is  6.  940  847 

Thus  the  logarithmic  sine  and  cosine  of  all  arcs  are  found.  After 
the  logarithmic  sine  and  cosine  of  any  arc  have  been  found,  the 
tangent  and  cotangent  of  the  same  arc  can  be  found  by  equations 
(3)  and  (5),  and  the  secants  by  (4);  that  is, 

R  sin.  a                     R  cos.  a                       R* 
tan.  a= cot.  a= sec.  a= 


cos  a  sin.  a  cos.  a 


62  SURVEYING. 

For  example,  the  logarithmic  sine  of  6P  is  9.019235,  and  its 
cosine  9.997614.  From  these,  find  tan.,  cot.,  and  secant. 

jRsin. 19.019235 

Cos.      -                 -               subtract    9.997614 
Tan.  is 9.021621 

jRcos. 19.997614 

Sin.            ...          subtract  9.019235 

Cotan.  is  10.978379 

R*  is 20.000000 

Cos.      -                  -         -      subtract  9.997674 

Secant  is    -                                     -  10.002326 
Thus  we  find  all  the  materials  for 

TABLE    II. 

This  table  contains  logarithmic  sines  and  tangents,  and  natural 
sines  and  cosines.  We  shall  confine  our  explanations  to  the  loga- 
rithmic shies  and  cosines. 

The  sine  of  every  degree  and  minute  of  the  quadrant  is  given, 
directly,  hi  the  table,  commencing  at  0°  and  extending  to  45°,  at 
the  head  of  the  table ;  and  from  45°  to  90°,  at  the  foot  of  the  table, 
increasing  backward. 

The  same  column  that  is  marked  sine  at  the  top,  is  marked  cosine 
at  the  bottom  ;  and  the  reason  for  this  is  apparent  to  any  one  who 
has  examined  the  definitions  of  sines. 

The  difference  of  two  consecutive  logarithms  is  given,  correspond- 
ing to  ten  seconds.      Removing  the  decimal  point  one  figure  will 
give  the  difference  for  one  second ;  and  if  we  multiply  this  difference 
by  any  proposed  number  of  seconds,  we  shall  have  a  difference 
corresponding  to  that  number  of  seconds,  above  the  logarithm, 
corresponding  to  the  preceding  degree  and  minute. 
For  example,  find  the  sine  of  19°  17'  22". 
The  sine  of  19°  17',  taken  directly  from  the  table,  is  9.518829 
The  difference  for  10"  is  60.2  ;  for  1",  is  6.02x22  133 

Hence,  19°  17'  22"  sine  is  9.518952 

From  this  it  will  be  perceived  that  there  is  no  difficulty  in  obtaining 
the  sin.  or  tan.,  cos.  or  cot.,  of  any  angle  greater  than  307. 


PLANE    TRIGONOMETRY.  63 

Conversely.  Given  the  logarithmic  sine  9.982412,  to  find  its  corres- 
ponding arc.  The  sine  next  less  in  the  table,  is  9.982404,  and  gives 
the  arc  73°  48'.  The  difference  between  this  and  the  given  sine,  is  8, 
and  the  difference  for  1",  is  .61  ;  therefore,  the  number  of  seconds  cor- 
responding to  8,  must  be  discovered  by  dividing  8  by  the  decimal  .61, 
which  gives  13.  Hence,  the  arc  sought  is  73°  48'  13". 

These  operations  are  too  obvious  to  require  a  rule.  When  the  arc 
is  very  small,  such  arcs  as  are  sometimes  required  in  astronomy,  it  is 
necessary  to  be  very  accurate  ;  and  for  that  reason  we  omitted  the 
difference  for  seconds  for  all  arcs  under  30'.  Assuming  that  the  sines 
and  tangents  of  arcs  under  30'  vary  in  the  same  proportion  as  the  arcs 
themselves,  we  can  find  the  sine  or  tangent  of  any  very  small  arc  to 
great  accuracy,  as  follows : 

The  sine  of  1',  as  expressed  in  the  table,  is  .  .  6.463726 
Divide  this  by  60 ;  that  is,  subtract  logarithm  .  .  1.778151 
The  logarithmic  sine  of  1",  therefore,  is  .  .  .  4.685575 
Now,  for  the  sine  of  17",  add  the  logarithm  of  17  .  1.230449 

Logarithrpic  sine  of  17",  is 5.916024 

In  the  same  manner  we  may  find  the  sine  of  any  other  small  arc. 

For  example,  find  the  sine  of  14'  2l£";  that  is,  861"5 

To  logarithmic  sine  of  1",  is,     .....      4.685575 

Add  logarithm  of  861.5 2.935255 

Logarithmic  sine  of  14'  2l£" 7.620830 

Without  further  preliminaries,  we  may  now  preceed  to  practical 

EXAMPLES. 

2.  In  a  right  angled  triangle,  ASC,  given 
the  base,  AS,  1214,  and  the  angle  A,  51°  40' 
30",  to  find  the  other  parts. 

To  find  BC. 

As  radius       .        .       10.000000 

:    tan.A  51°  40' 30"  10.102119 

::   AB  1214         .        3.084219 

:    BC  1535.8      .        3.186338 

N.  B.  When  the  first  term  of  a  logarithmic  proportion  is  radius, 
the  resulting  logarithm  is  found  by  adding  the  second  and  third  loga- 
rithms, rejecting  10  in  the  index,  which  is  dividing  by  the  first  term. 

In  all  cases  we  add  the  second  and  third  logarithms  together;  which, 
in  logarithms,  is  multiplying  these  terms  together;  and  from  that  sum 


64  SURVEYING. 

we   subtract  the  first  logarithm,  whatever  it  may  be,  which  is 
dividing  by  the  first  term. 

To  find  A  0. 

As  sin.  (7,  or  cos.  A  51°  30'  40"    -  9.792477 

:  AE12U  -     3.084219 

:  :  Badius         -         -        -        10.000000 


:  .4(71957.7       -         -         -     3.291742 

To  find  this  resulting  logarithm,  we  subtracted  the  first  logarithm 
from  the  second,  conceiving  its  index  to  be  13. 

Let  AB  C  represent  any  plane  triangle,  right  angled  at  JB. 

1.  Given  .4(773.26,  and  the  angle  A  49°  12'  20"  ;  required  the 
other  parts. 

Ans.  The  angle  C  40°  47'  40",  5(755.46,  and  AB  47.87. 

2.  Given  AB  469.34,  and  the  angle  A  51°  26'  17",  to  find  the 
other  parts. 

Ans.  The  angle  (7  38°  33'  43",  BO  588.5,  and  A  0  752.9. 

3.  Given  AB  493,  and  the  angle  (720°  14';  required  the  remain- 
ing parts.          Ans.  The  angle  A  69°  46',  B  C  1 338,  and  A  0  1 425. 

It  is  not  necessary  to  give  any  more  examples  in  right  angled 
plane  trigonometry,  for  every  distance  in  the  traverse  table  is  but  the 
hypotenuse  of  a  right  angled  triangle,  and  its  corresponding  latitude 
and  departure  form  the  sides  of  the  triangle. 

If  any  one  should  suspect  an  error  in  the  traverse  table,  let  him  test  it 
ty  computing  the  triangle  anew. 

OBLIQUE    ANGLED    TRIGONOMETRY. 

Of  the  six  parts  of  a  triangle,  three  sides  and  three  angles,  three 
of  them  must  be  given  and  one  of  the  given  parts  must  be  a  side. 
The  subject  presents  four  cases. 

1.  When  two  sides  are  given,  and  an  angle  opposite  one  of  them. 

2.  When  two  sides  are  given,  and  the  included  angle. 

3.  When  one  side  and  two  angles  are  given. 

4.  When  the  three  sides  are  given. 

The  principles  previously  demonstrated  are  sufficient,  and  indeed 
ample,  to  give  all  solutions  that  can  come  under  any  one  of  these 


OBLIQUE    ANGLED    TRIGONOMETRY.  65 

cases.     The  operator  must  use  his  own  judgment  in  applying  these 
principles. 

We  give  an  example  in  each  case,  which,  with  the  incidental 
examples,  will  be  sufficient  to  fix  the  principles  in  the  mind  of  the 
operator. 

EXAMPLE    1. 

In  any  plane  triangle,  given  one  side  and  the  two  adjacent  angles ,  to 
find  the  other  sides  and  angle. 

In  the  triangle  ABC,  given  JLB=376,  the  angle  -4=48°  3',  and 
the  angle  .5=40°  14',  to  find  the  other  parts. 

As  the  sum  of  the  three  angles  of  every  triangle  is  equal  to  180°, 
the  third  angle  C  must  be  180°— 88°  17'=91°  43'. 

INSTRUMENTALITY. 

Take  376  from  the  scale,  by  means  of  the  dividers,  and  place  it 
on  paper;  making  one  extremity  of  the  line  A,  and  the  other 
extremity  B.  From  A,  by  means  of  the  protractor  (or  otherwise), 
make  the  angle  ^4=48°  3',  and  from  B,  make  the  angle 
.£=40°  14'.  The  intersections  of  the  lines  AC,  EG,  will  give  the 
angle  C,  which  being  measured  will  be  found  to  be  a  little  more 
than  a  right  angle. 

Take  A  C  in  the  dividers,  and  apply  it  to  the  scale,  and  it  will  be 
found  to  be  243  ;  and  BG  will  be  found  to  be  279.8,  if  the  projec- 
tion is  accurately  made ;  but  no  one  should  expect  numerical  accuracy 
from  this  mechanical  method. 

N.  B.  Our  figures  in  the  book  do  not  pretend  to  accuracy,  they  should  be 
drawn  on  paper  on  a  larger  scale. 

BY     LOGARITHMS. 

To  find  A  C. 

As  sin.  91°  43'  -  9.999805 

:    AB  376  ...  2.575188 

::  sin.  AB  40°  14'     -         -        -  9.810167 


12.385355 
:   ^1(7243       -         -  2.385550 

Observe,  that  the  sine  of  91°  43'  is  the  same  as  the  cosine  of  1  °  43* 
5 


66  SURVEYING. 

To  find  EC. 

As  sin.  91°  43'   -         -        -  9.999805 

:   AB376  -        -        -        -     2.575188 

:  :  sin.  ^148°  3'  -        -         9.871414 

12.446602 


:   £(7279.8      ....        2.446797 
EXAMPLE    2, 

In  a  plane  triangle,  given  two  sides,  and  an  angle  opposite  one  of 
them,  to  determine  the  other  parts. 

Let  AD=  1751 .  feet,  one  of  the  given 
sides.  The  angle  D=31°  17'  19",  and 
the  side  opposite,  1257.5.  From  these 
data,  we  are  required  to  find  the  other 
side,  and  the  other  two  angles. 

In  this  case  we  do  not  know  whether 
A  C  or  AE  represents  1 257.5,  because 
A  C=AE.  If  we  take  A  C  for  the  other  given  side,  then  DO  is  the 
other  required  side,  and  DA  C  is  the  vertical  angle.  If  we  take  AE 
for  the  other  given  side,  then  DE  is  the  required  side,  and  DAE  is 
the  vertical  angle ;  but  in  such  cases  we  determine  both  triangles. 

INSTRUMENTALITY. 

Draw  DE  indefinitely — from  the  point  D  make  the  angle 
Z>=31°17'.  AD=1751.,  but  call  it  175.1,  which  take  from  the 
scale.  Place  one  foot  of  the  dividers  at  D,  the  other  foot  will  extend 
to  Ay  thus  finding  the  point  A. 

Take  125.75  in  the  dividers,  place  one  foot  at  A  as  a  center,  and 
with  the  other  strike  an  arc,  cutting  DE  in  C  and  E.  Join  A  C, 
AE,  and  one  or  the  other  of  the  triangles  A  CD  ADE,  will  be  the 
triangle  required.  D  0  and  DE  applied  to  the  scale,  will  give  one- 
tenth  of  the  required  side,  and  the  angle  E  or  DC  A,  measured,  will 
be  one  of  the  required  angles. 

We  can  also  take  one  hundredth  part  of  the  sides,  as  well  as  the 
tenth  ;  this  will  make  no  difference  with  the  angles,  the  triangles 
thus  formed  will  be  similar. 

In  that  case  AD— 17.51,  and  the  side  sought  will  be  23.64,  which 
can  be  changed  to  2364. 


V. 

OF 

:  srr 


OBLIQUE    ANGLED    TRIGONOMETRY.          67 

BY     LOGARITHMS. 

To  find  the  angle  £=C. 

(Prop.  4.)  As  A  C=Afi=1257.5    log.  3.099508 

:  Z>31°  17'  19"  sin.  9.715460 

:  :  AD  1761  log.  3.243286 

12.958746 


j^=  (7 :  46°  18'  sin.  9.859238 

From  180°  take  46°  18',  and  the  remainder  is  the  angle  DC  A 
=  133°  42'. 

The  angle  DAC=ACE—D (th.  1 1,  b.  1)  ;  that  is, 
DAC=46°  18'— 31°  17'  19"=15°0'  41". 

The  angles  D  and  E,  taken  from  180°,  give  DAE=W2°  24' 41". 

To  findJ)  0. 

As  sin.  D  31°  17'  19"  log.     9.715460 

:   -4(71257.5  log.     3.099508 

:  :  sin.  DAG  15°  0'41"  log.     9.413317 

12.512825 


:   JK7626.86  2.797165 

To  find  DE. 

As  sin.  D  31°  17'  19  9.715460 

:  .4^1257.5  3.099508 

:  :  sin.  102°  24'  41"  9.989730 


13.089238 
:    DE  2364.5  3.373778 

N.  B.  To  make  the  triangle  possible,  A  C  must  not  be  less  than 
AB,  the  sine  of  the  angle  D,  when  DA  is  made  radius. 

EXAMPLE    3. 

In  any  plane  triangle,  given  two  sides  and  the  included  angle,  to  find 
the  other  parts. 

Let  AZ>=1751  (see  last  figure),  7)^=2364.5,  and  the  included 
angle  D=41°  17'  19".  We  are  required  to  find  DE,  the  angle 
DAE,  and  angle  E.  Observe  that  the  angle  E  must  be  less  than 
the  angle  DAE,  because  it  is  opposite  a  less  side. 


68  SURVEYING. 

INSTRUMENTALLY. 

Take  D.#=236.45  from  the  scale  (as  near  as  possible),  and  from 
&  draw  J)A,  making  the  given  angle  31°  17'  19". 

Take  175.1  from  the  scale,  in  the  dividers,  and  with  it  mark  off 
DA.  Join  AE ;  and  ADE  will  be  the  triangle  in  question,  and  AE 
applied  to  the  scale  will  give  the  tenth  part  of  the  side  sought ;  and 
measuring  the  angle  E  with  the  protractor  (or  otherwise),  will 
determine  its  value. 

BY    LOGARITHMS. 

From  -  -       180° 

Take  D         -         -         -         -    31°  17'  19" 


Sum  of  the  other  two  angles  =148°  42'  41"  (th.  11,  b.  1) 
4  sum  -         -  =  74°  21' 20" 

By  proposition  7, 

DE+DA  :  DE—DA—i^n.  74°  21'  20"  :  tan.  |  (DAE—E) 
That  is, 

4115.5  :  613.5=tan.  74°  21'  20"  :  l^DAE—E) 

Tan.  74°  21'  20"  10.552778 

613.5  -  -  2.787815 

13.340593 

4115.5  log.  (sub.)  3.614423 

|(  DAE—  E)  tan.  28°  1 '  36"  9.726170 

But  the  half  sum  and  half  difference  of  any  two  quantities  are 
equal  to  the  greater  of  the  two  ;  and  the  half  sum,  less  the  half 
difference,  is  equal  the  less. 

Therefore,  to    74°  21' 20" 
Add  28      1  36 


102°  22' 56" 
jE=  46    19  44 
To  find  AE. 

As  sin.  ^46°  19'  44"  9.859323 

:   J9^1751  -      3.243286 

:  :  sin.  D  31°  17' 19"         -        -          9.715460 


12.958746 
AE  1257.2          -        -        -  3.099423 


OBLIQUE    ANGLED    TRIGONOMETRY.         69 

EXAMPLE    4. 
Given  the  three  sides  of  a  plane  triangle  to  find  the  angles. 

Given  AC=  1751,  (75=1257.5, -4 JS= 2364.5 
If  we  take  the  formula  for  cosines,  we  will 
compute  the  greatest  angle,  which  is  0. 


INSTRUMENTALITY. 

Construct  a  triangle  with  the  three  given 
sides  236.45,  125.75,  and  175.1,  according 
to  problem  16,  chapter  1.     The  angles  then  measured  will  show 
their  value. 

BT     LOGARITHMS . 

R*  20.000000 

5=2686.5  3.429187 

=322  2.507856 


Numerator,  log.         25.937043 

a  1257.5  3.099508 

51751.  3.243286 


Denominator,  log.  6.342794      6.342694 

2)19.594249 

|(7=  51°  ll'10"cos.  9.797124 

(7=102    22  20 

The  remaining  angles  may  now  be  found  by  problem  4. 
We  give  the  following  examples  for  practical  exercises : 
Let  ABC  represent  any  oblique  angled  triangle. 

1.  Given  ^LB697,  the  angle  A  81°  30' 10",  and  the  angle  B  40° 
30'  44",  to  find  the  other  parts. 

Ans.  AC 534,  £(7813,  and  the  angle  (757°  59' 4". 

2.  If  .4(7=720.8,  the  angle  .4=70°  5'  22",  and  the  angle  B= 
59°  35'  36",  required  the  other  parts. 

Ans.  AB  643.2,  5(7785,8,  and  the  angle  (750°  19'  6". 

3.  Given  5(7980.1,  the  angle  A  7°  6'  26",  and  the  angle  B 106° 
2'  23",  to  find  the  other  parts. 

Ans.  AB  7284,  .4(77613.3,  and  the  angle  (766°  51'  11". 


SUKVEYING. 

is  the  art  of  running  definite  lines  on  the  surface  of  the 
earth,  measuring  them,  and  finding  the  contents  of  lands  ;  and  the 
subject  necessarily  includes  the  measure  of  surfaces  generally.  We 
shall  therefore  commence  with  mensuration. 

Mensuration  is  the  application  of  the  principles  of  Geometry,  to 
the  measure  of  surfaces  and  solids,  and  when  lands  are  measured  it 
is  a  part  of  surveying.  We  shall  be  very  brief  on  mensuration 
proper,  because  the  rules  are  so  simple  and  obvious.  For  the 
demonstration  of  the  rules,  we  refer  to  (Legendre  and  Robinson's 
Geometry.) 

All  surfaces  are  measured  by  the  number  of  square  units  which 
they  contain.  The  unit  may  be  taken  at  pleasure ;  it  may  be  an 
inch,  foot,  yard,  rod,  mile,  &c.,  as  convenience  and  propriety  may 
dictate. 

The  square  unit  is  always  the  square  of  the  linear  unit. 

PROBLEM   I. 

To  find  the  area  of  a  square,  or  a  parallelogram. 

RULE. —  Multiply  the  length  by  the  perpendicular  breadth,  and  the 
product  will  be  the  area. 

(Leg.  b.  IV,  prop.  V.    Rob.  book  I,  th.  29). 

1.  What  is  the  area  of  a  square  whose  sides  are  6  feet  3  inches  ? 

Ans.  39  Jj  square  feet.  * 

2.  How  many  square  feet  are  in  a  board  that  is  13^  feet  long 
and  10  inches  wide?  Ans.  11£  square  feet. 

3.  A  lot  of  land  is  80  rods  long,  and  45  rods  wide,  how  many 
square  rods  does  it  contain,  and  how  many  acres  ? 

Ans.  3600  rods,  22£  acres. 

*  NOTE. —  Reductions  from  one  measure  to  another  have  no  reference  to  the 
rales  here  given. 
(70) 


MENSURATION.  71 

4.  A  man  bought  a  farm  198  rods  long,  and  150  rods  wide,  at 
$32  per  acre ;  what  did  it  come  to  ?  Ans.  $5940. 

PROBLEM  II. 

To  find  the  area  of  a  triangle,  when  the  base  and  altitude  are 
given. 

RULE. —  Multiply  one  of  these  dimensions  by  half  the  other,  and 
the  product  will  be  the  area  required. 

(Leg.  book  IV,  p.  VI.    Rob.  book  I,  th.  30). 

1.  How  many  yards  hi  a  triangle  whose  base  is  148  feet,  and 
perpendicular  45  feet  ?  Ans.  370  yards. 

2.  What  is  the  area  of  a  triangle  whose  base  is  18i  feet  and  alti- 
tude 251  feet  ?  Ans.  231£i  feet. 

PROBLEM  III. 

Investigate  and  give  a  rule  for  finding  the  area  of  a  triangle  when 
two  sides  and  their  included  angle  are  given. 

Let  AEG  be  the  triangles,  AB,  EG  the 
given  sides,  and  E  the  given  angle. 

Represent  the  side  opposite  to  the  angle  A, 
by  a,  opposite  (7,  by  c,  and  opposite  B,  by  b. 

Now  a  and  c  are  the  given  sides,  and  by  problem  II,  the  area  is 

WAD)  (i) 

The  trigometrical  value  of  AD  can  be  found  from  the  right 
angled  triangle  ABD. 

Thus,  sin.  ADB  :  c  :  :  sin.  B  :  AD. 

That  is,  1  :  c  :  :  sin.  B  :  AD. 

Whence  AD=c  sin.  B. 

This  value  of  AD  substituted  in  ( 1 )  gives 

\ac  sin.  £=  area  A  (2). 

This  expression  is  the  area  of  the  triangle,  and  from  it  we  draw 
the  following  rule. 

RULE. —  Take  half  the  product  of  the  two  given  sides  and  multiply 
it  by  the  natural  sine  of  the  included  angle,  and  the  last  product  will  be 


72  SURVEYING. 

1.  One  side  of  a  triangle  is  84  feet,  another  90  feet,  and  their 
included  angle  is  27°  31'.    What  is  the  area  ? 

Ans.  1746.4  square  feet. 
27°  31'  Nat.  sine.         -  -    46201 

iac 3780 

3696080 
323407 
138603 

1746.39780  Ans. 

When  we  use  logarithms  we  have  the  following  rule  : 
RULE. —  To  the  logarithms  of  the  two  sides,  add  the  log.  sine  of  the 

included  angle,  and  the  sum  rejecting  10,  in  the  index,  is  the  logarithm 

of  twice  the  area  of  the  triangle. 

2.  A  certain  triangle  has  one  side  125.81,  another  equal  57.65, 
and  their  included  angle  57°  25',  what  is  its  area  ?         Ans.  3055.7. 

125.81      log.  2.099715 

57.65      log.  1.760799 

57°  25'  sine  9.925626 


2  Area,     6111.4    log.  3.786140 

3.  How  many  square  yards  hi  a  triangle,  two  sides  of  which  are 
25  and  21J  feet,  and  their  included  angle  45°  ?  Ans.  20.8695. 

PROBLEM   IV. 

Investigate  and  give  a  rule  for  finding  the  area  of  a  triangle  when 
the  three  sides  are  given. 

(See  figure  to  problem  III).  Let  A  represent  the  area  of  any 
plane  triangle,  then  by  problem  III 

A=\ac$\a.B.  (1) 

But  sin.  .#=2  sin.  \B  cos.  %B.     (Eq.  30,  trigonometry). 

Therefore,        A=ac,  sin.  \B  cos.  \E.  (2) 

Now  in  proposition  8,  trigonometry,  we  find 

(3) 
and      cos.i.B=v/!i£l±2  (4) 


PLANE    TRIGONOMETRY.  73 

The  product  of  (3)  into  (4)  is 


sin.  LB  cos.  ^===,  (5) 


or         ac  sin.  £2?  cos.  ±B=  J  s(s—a)(s — b)(s—c).      (6) 
By  comparing  (2)  and  (6)  we  perceive  that 
A=,Js(s— a)(s—b)(s—c). 

Here  s  represents  the  half  sum  of  a,  b,  and  c,  therefore,  we  have 
the  following  rule  to  find  the  area  when  the  three  sides  are  given. 

RULE. —  Add  the  three  sides  togettier  and  take  half  the  sum.  From 
the  half  sum  take  each  side  separately,  thus  obtaining  three  remainders. 
Multiply  the  said  half  sum  and  the  three  remainders  together ;  the 
square  root  of  this  product  is  the  area  required. 

1.  Find  the  area  of  a  triangle  whose  sides  are  20,  30,  and  40. 

Ans.  290.47. 
i  sum  =45,   1st  Hem.  =25,  2d=15,  3d=5. 


745.25.15.5=7225  25.15=15.5,/15=:75(3.873)  =  290.474. 

2.  How  many  acres  in  a  triangle  whose  sides  are  severally  60, 
50,  and  40  rods  ?  Ans.  6^  nearly. 

3.  How  many  square  yards  are  there  in  a  triangle  whose  sides 
are  30,  40  and  50  feet  ?  Ans.  66=-. 

4.  There  is  a  triangular  lot  of  land  containing  8  acres,  two  of 
its  sides  are  64,  and  46  rods  respectively ;  what  is  the  angle  be- 
tween these  sides,  and  what  is  the  length  of  the  remaining  side  ? 

Ans.     The  angle  is  60°  25',  or  is  supplement  119°  35'. 
The  side  is  57.37  rods,  or  95.535  rods ;    the  less  angle  corres- 
ponding to  the  lesser  side. 

In  short,  there  are  two  triangles  answering 
to  the  conditions,  the  one  is  ABE,  the  other 
ABC.  They  are  equal  because  they  are  on 
the  same  base  and  between  the  same  parallels. 
AE  =  57.37,  AC  =  95.535. 
7 


74  SURVEYING. 

PROBLEM  V. 

To  find  the  area  of  a  trapezoid. 

RULE. —  Add  the  two  parallel  sides  together,  and  take  half  the  sum. 
Multiply  this  half  sum  by  the  perpendicular  distance  between  the  sides. 

Or,  The  sum  of  the  parallel  sides  multiplied  by  their  distance 
asunder  will  give  twice  the  area. 

(Leg.  book  IV,  prop.  VII.     Rob.  b.  I,  th.  31). 

REMARK. —  The  application  of  this  problem  is  the  most  important  of  any 
in  general  surveying,  as  will  appear  in  the  sequel,  and  if  the  geometrical 
theorem  is  not  familiar  to  the  student  he  should  again  review  it 

Ex.  1.  In  a  trapezoid,  the  parallel  sides  are  750  and  1225,  and 
the  perpendicular  distance  between  them  1 540  links :  to  find  the 
area. 

1225 
750 

1975X770=152075  square  links  =15  acr.  33  perches. 
Ex.  2.  How  many  square  feet  are  contained  in  a  plank,  whose 
length  is   12  feet  six  inches,  the  breadth  at  the  greater  end   15 
inches,  and  at  the  less  end  11  inches?  Ans.  13^J  feet. 

Ex.  3.  In  measuring  along  one  side  AB  of  a  quadrangular  field, 
that  side,  and  the  two  perpendiculars  let  fall  on  it  from  the  two 
opposite  corners,  measured  as  below,  required  the  content. 
AP  =     110  links 
AQ  =    745 
AB  =  1110 
CP=    352 
DQ  =    595 

Ans.  4  acres,  1  rood,  5.792  perches. 

Here  we  perceive  a  trapezoid  and  two  right  angled  triangles. 
N.  B.     A  chain  is  4  rods,  and  contains  100  links ;    10  square 
chains  make  an  acre. 

PROBLEM  VI. 

To  find  the  Area  of  any  Trapezium. 

DIVIDE  the  trapezium  into  two  triangles  by  a  diagonal ;  then  find 
the  areas  of  these  triangles,  and  add  them  together. 


MENSURATION  75 

Or  thus,  let  fall  two  perpendiculars  on  the  diagonal  from  the  other 
two  opposite  angles  ;  then  add  these  two  perpendiculars  together, 
and  multiply  that  sum  by  the  diagonal,  taking  half  the  product  for 
the  area  of  the  trapezium. 

Ex.  1  .  To  find  the  area  of  the  trapezium,  whose  diagonal  is  42, 
and  the  two  perpendiculars  on  it  16  and  18. 
Here  16-fl8=    34,  its  half  is  17. 
Then  42  X  17=  714  the  area. 

Ex.  2.  How  many  square  yards  of  paving  are  in  the  trapezium, 
whose  diagonal  is  65  feet  ;  and  the  two  perpendiculars  let  fall  on  it 
28  and  33^  feet  ?  Ans.  222^  yards. 

When  the  sides  of  a  trapezium,  and  two  of  its  opposite  angles  are 
given,  the  most  convenient  rule  for  finding  its  area  is  found  in 
problem  III. 

Conceive  CB  joined,  then  the  whole  figure 
consists  of  two  triangles  and  the  whole  area 
is  found  in  the  following  expression 
(AB  X  ACX  sm.A)  -f-  (CD  X  DB  X  sin. 
CDB.) 

EXAMPLE. 

In  the  quadrilateral  ACDB  we  have  AC  15.7,  CD  20.4,  DB 
14.24,  and  BA  27.7  rods.  The  angle  A  78°  15'  and  the  opposite 
angle  CDB  97°  30'.  What  is  the  area  enclosed  ? 

Ans.  356.65  square  rods. 

PROBLEM   VII. 

To  find  the  area  ©f  an  irregular  figure  bounded  by  any  number 
of  right  lines. 

RULE.  —  Draw  diagonals  dividing  the  figure  into  triangles.  Find 
the  areas  of  the  triangles  so  formed  and  add  them  together  for  the  area 
of  the  whole. 

Let  it  be  required  to  find  the 
area  of  the  adjoining  figure  of 
five  sides.  On  the  supposition 
that  A  (7=36.21  .#(7=39.11 


Ans.  296.129. 


76  SURVEYING. 

PROBLEM    VIII. 

To  find  the  area  of  a  long  irregular 
figure  like  the  one  represented  in  the 
margin,  it  is  necessary  to  divide  it  into 
trapezoids.  Then  find  the  area  of  each 
one  of  the  trapezoids  (by  problem  V.)  and  add  them  together  for  the 
whole  area. 

If  however  the  trapezoids  have  equal  distances  between  their 
parallel  sides  we  can  take  a  more  summary  process,  which  we  dis- 
cover by  the  following  investigation. 

The  trapezoid  AEFD=\  (a+b)XAE. 

GI£H  =  l(c4-d)X0I. 


On  the  supposition  that  AE,  EG,  GI,  &c.  are  all  equal  to  each 
other  the  sum  of  these  is  / 


which  represents  the  area  of  the  whole  figure. 

From  this  we  draw  the  following  rule  to  find  the  area  of  a  long 

and  narrow  figure  bounded  by  a  right  line  on  one  side,  and  a  broken 

or  curve  line  on  the  other,  to  which  off  sets  are  made  at  equidistant 

points  along  the  right  line. 

RULE.  —  Add  the  intermediate  breadths  or  offsets  together,  and  the  half 

sum  of  the  extreme  one  :  then  multiply  this  sum  by  one  of  the  equal  parts 

of  the  right  line,  the  product  will  be  the  area  required,  very  nearly  * 
1.  The  breadths  of  an  irregular  figure  at  five  equidistant  places, 

being  6.2,  5.4,  9.2,  3.1,  4.2,  and  the  length  of  the  base  60,  what  is 

the  area  ? 

Mean  of  the  Extremes  5.2 

Sum  of  5.4,  9.2,  3.1  17.7 

Sum  22.9 

One  of  the  equal  parts  1  5 

1145 
229 
Area==343.5 

»  In  case  DF,  FH,  &c.  are  right  lines  we  shall  have  the  area  exactly,  if  they 
are  other  than  right  lines  the  area  will  be  nearly. 


MENSURATION  77 

2  The  length  of  an  irregular  figure  being  84,  and  the  breadths 
at  oix  equidistant  places  17.4  20.6  14.2  16.5  20.1  24.4  ;  what 
is  the  area?  Ans.  1550.64. 

PROBLEM    IX. 

To  find  the  area  of  a  circle,  also  any  sector  or  segment  of  a 
circle. 

RULE  1 .  —  The  area  of  a  circle  is  found  by  multiplying  the  radius 
by  half  the  circumference. 

(Leg.  book  V,  prop.  12.     Rob.  book  V,  th.  1.) 

RULE  2.  Multiply  the  square  of  the  diameter  by  the  decimal  .7854. 

When  the  radius  of  a  circle  is  1,  the  length  of  one  degree  on  the 
circumference  is  0.01745  and  the  whole  circumference  is  3.1416. 

The  radius  and  the  circumference  increase  and  decrease  by  the 
same  ratio,  therefore  the  length  of  any  arc  corresponding  to  any 
radius  is  easily  computed. 

A  sector  of  a  circle  is  to  the  whole  circle  as  the  number  of  degrees 
it  contains  is  to  360. 

The  area  of  a  segment  of  a  circle  as  FAE, 
may  be  found  by  first  finding  the  sector  FCE, 
and  from  it  taking  the  area  of  the  triangle 
FCE. 

This  same  triangle  added  to  the  greater  sec- 
tor will  give  the  greater  segment. 

These  principles  and  rules  are  sufficient  to 
solve  the  following  examples  which  are  given  merely  as  educational 
Exercises. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  10  ? 

Ans.  78.54. 

2.  What  is  the  area  of  a  circle  whose  diameter  is  20  ? 

Ans.  4  times  78.54. 

3.  What  is  the  area  of  a  circle  whose  circumference  is  12  ? 

Ans.  11.4595. 

4.  How  many  square  yards  are  in  a  circle  whose  diameter  is  3|- 
feet?  Ans.  1.069. 

5.  Find  the  length  of  an  arc  of  20°,  the  radius  being  9  feet. 

Ans.  3.141. 


78  SURVEYING. 

6.  Find  the  length  of  an  arc  of  60°,  the  radius  being  18  feet. 

Ans.  18.846 

7.  To  find  the  length  of  an  arc  of  30  degrees,  the  radius  being  9 
feet-  Ans.  4.7115, 

8.  To  find  the  length  of  an  arc  of  12°  10',  or  12%  the  radius 
being  10  feet.  Ans.  2.1231. 

9.  What  is  the  area  of  a  circular  sector  whose  arc  is  1 8°  and  the 
diameter  3  feet  ?  Ans.  0.35343. 

10.  To  find  the  area  of  a  sector,  whose  radius  is  10,  and  arc  20. 

Ans.  100. 

11.  Required  the  area  of  a  sector,  whose  radius  is  25,  and  its 
arc  containing  147°  29'.  Ans.  804.3986. 

12.  What  is  the  area  of  the  segment,  whose  height  is  18,  and 
diameter  of  the  circle  60  ?  Ans.  636.375. 

13.  Required  the  area  of  the  segment  whose  chord  is  16,  the 
diameter  being  20  ?  Ans.  44.728. 

14.  What  is  the  length  of  a  chord  which  cuts  off  one-third  of 
the  area  from  a  circle  whose  diameter  is  289  ?  Ans.  278.6716. 

15.  The  radius  of  a  certain  circle  is  10 ;  what  is  the  area  of  a 
segment  whose  chord  is  12?  Ans.  16.35. 

16.  What  is  the  area  of  a  segment  whose  height  is  2  and  chord 
20?  Ans.  26.88. 

17.  What  is  the  area  of  a  segment  whose  height  is  5,  the  diame- 
ter of  the  circle  being  8  ?  Ans.  33.0486. 

PROBLEM  X. 

To  find  the  Area  of  an  Ellipse. 

RULE. —  Multiply  the  two  semi-axes  togetJier  and  their  product  by 
3.1416.  (See  conic  sections). 

1.  Required  the  area  of  an  Ellipse  whose  two  semi-axes  are  25 
and  20.  Ans.  1570.8. 

2.  The  two  semi-axes  of  an  Ellipse  are  12  and  9,  what  is  its  area  ? 

Ans.  339.29. 

To  find  the  area  of  any  portion  of  a  parabola  we  multiply  the 
base  by  the  perpendicular  height,  and  take  two-thirds  of  the  product  for 
the  area  required.  (See  conic  sections). 


MENSURATION    OF   SOLIDS.  79 

Required  the  area  of  a  parabola,  the  base  being  20,  and  the  alti- 
tude 30.  Ans.  400. 

The  surfaces  of  prisms,  cylinders,  pyramids,  cones,  &c.,  are  found 
by  the  application  of  the  preceding  rules. 

From  theorem  16,  book  VII,  Geometry,  we  learn  that 

The  convex  surface  of  a  sphere  is  equal  to  the  product  of  its  diame- 
ter into  its  circumference. 

The  surface  of  a  segment  is  equal  to  the  circumference  of  the  sphere t 
multiplied  into  the  thickness  of  the  segment. 

In  the  same  sphere,  or  in  equal  spheres,  the  surfaces  of  different  seg- 
ments are  to  each  other  as  their  altitudes. 

MENSURATION  OF  SOLIDS. 

BY  the  Mensuration  of  Solids  are  determined  the  spaces  included 
by  contiguous  surfaces  ;  and  the  sum  of  the  measures  of  these 
including  surfaces,  is  the  whole  surface  or  superficies  of  the  body. 

The  measures  of  a  solid,  is  called  its  solidity,  capacity,  or 
content. 

Solids  are  measured  by  cubes,  whose  sides  are  inches,  or  feet,  or 
yards,  &c.  And  hence  the  solidity  of  a  body  is  said  to  be  so  many 
cubic  niches,  feet,  yards,  &c.,  as  will  fill  its  capacity  or  space,  or 
another  of  an  equal  magnitude. 

The  least  solid  measure  is  the  cubic  inch,  other  cubes  being  taken 
from  it  according  to  the  proportion  in  the  following  table,  which  is 
formed  by  cubing  the  linear  proportions. 

Table  of  Cubic  or  Solid  Measures. 


1728     cubic  inches  make 
27     cubic  feet 


cubic  yards 
64000     cubic  poles 
512     cubic  furlongs 


cubic  foot 
cubic  yard 
cubic  pole 
cubic  furlong 
cubic  mile. 


As  the  mensuration  of  solids  has  little  to  do  with  surveying  or 
navigation,  we  shall  leave  this  subject  after  simply  stating  the  fol- 
lowing truths,  which  are  demonstrated  in  solid  geometry. 

In  fact,  these  truths  may  be  called  rules  for  practical  operations. 


80  SURVEYING. 

1.  The  solidity  of  a  cube,  parallelepiped,  prism,  or  cylinder,  is 
found  by  multiplying  the  area  of  Us  base  by  the  altitude. 

2.  The  solidity  of  a  pyramid  or  cone  is  found  by  multiplying  the 
base  by  the  altitude,  and  taking  one-third  of  the  product. 

3.  The  solidity  of  the  frustum  of  a  pyramid  or  cone  is  found  by 
calculating  the  solidity  of  the  pyramid  when   complete,  and  subtracting 
from  it  the  solidity  of  the  part  removed  ;  or  find  the  area  of  the  top 
and  bottom  of  the  frustum.,  and  the  mean  proportional  between  these 
two  areas.      Add  these  three  quantities  together,  and  multiply  the 
sum  by  one-third  of  the  altitude  of  the  frustum,  and  the  product 
mil  be  the  solidity  sought. 

4.  Guaging  is  performed  by  considering  a  cask  to  be  made  up  of 
two  frustums  of  cones  placed  base  to  base,  and  applying  the  rules  for 
the  measurement  of  such  solids. 

5.  The  solidity  of  a  sphere  is  two-thirds  of  the  solidity  of  its  cir- 
cumscribing cylinder. 


CHAPTER    I. 

MENSURATION    OF    LANDS. 

LANDS  are  not  only  measured  to  find  their  areas,  but  their  exact 
positions  must  be  ascertained,  the  direction  which  each  line  makes 
with  the  meridians,  or  with  the  north  and  south  lines  on  the  earth. 

The  boundaries  of  each  tract  of  land  are  referred  to  that  meri- 
dian which  runs  through  or  by  the  side  of  it. 

All  meridian  lines  meet  at  the  poles,  therefore  they  are  not 
parallel,  (except  at  the  equator,)  but  the  poles  are  so  far  distant 
that  no  sensible  error  can  arise  from  supposing  them  parallel,  and 
all  surveys  are  made  on  the  supposition  that  the  surface  of  the  earth 
is  a  plane  and  the  meridians  parallel.  When  large  surveys  are 
made,  like  a  county  or  a  state,  the  spherical  form  of  the  earth 
should  be  taken  into  consideration. 

Meridian  lines  in  surveys  are  usually  determined  by  the  magnetic 
needle,  but  the  needle  does  not  settle  exactly  north  and  south,  gen- 


MENSURATION   OF  LANDS.  81 

erally  speaking,  and  the  direction  which  it  does  settle  is  called  the 
magnetic  meridian. 

Surveys  are  often  made  by  the  magnetic  meridian  as  the  true  one, 
and  this  would  answer  every  purpose,  provided  the  difference  be- 
tween the  magnetic  and  true  meridians  were  every  where  and  at  all 
times  the  same,  but  this  is  not  so. 

The  magnetic  meridian  is  variable,  and  for  this  reason  it  is  very 
difficult  to  trace  old  lines,  unless  visible  monuments  are  left,  or 
unless  the  record  refers  to  the  true  meridian. 

Lines  are  generally  measured  by  a  chain  of  66  feet  or  4  rods  in 
length,  containing  100  links,  each  link  is  therefore  7.92  inches. 

The  area  of  land  is  estimated  in  acres  and  hundredths,  formerly 
in  acres,  roods,  and  perches,  but  the  modern  method  is  more  simple 
and  convenient ;  we  have  a  clearer  conception  of  35  hundreths  of 
an  acre  than  we  have  of  1  rood  and  16  perches. 

An  acre  is  equal  to  10  square  chains  or  100,000  square  links. 

We  may  note  down  the  length  of  a  line  in  chains  and  hundreths, 
or  in  links  only,  for  it  is  nearly  one  and  the  same  thing  :  thus,  12 
chains  and  38  links  may  be  written  12.38,  or  1238  links. 

The  area  of  a  field  may  be  found  by  measuring  with  the  chain 
only,  and  dividing  it  into  rectangles  and  triangles,  and  computing 
each  of  them  separately,  according  to  the  rules  laid  down  in  men- 
suration. 

The  most  common  method  for  measuring  a  field  for  calculation, 
is,  to  take  the  length  of  all  the  sides  of  the  field  with  the  chain,  and 
their  bearings  with  the  surveyor's  compass.  With  these  notes  an 
accurate  plan  or  plot  of  the  field  may  be  made  on  paper,  and  then 
its  contents  ascertained  by  cutting  it  into  triangles  and  measuring 
their  bases  and  perpendiculars  with  a  scale  and  dividers.  A  very 
little  instruction  from  a  teacher  will  enable  the  student  to  practice 
this  method  with  success  ;  yet  no  instrumental  measures  pretend  to  be 
numerically  accurate,  they  are  but  approximately  so. 

TO     MEASURE     A     LINE. 

Provide  a  chain  and  10  small  arrows  or  marking  pins  to  fix  one 
into  the  ground,  as  a  mark,  at  the  end  of  every  chain  ;  two  persons 

take  hold  of  the  chain,  one  at  each  end  of  it ;  and  all  the  10  arrows 
6 


82  SURVEYING. 

are  taken  by  one  of  them  who  goes  foremost,  and  is  called  t&* 
leader  ;  the  other  being  called  the  follower,  for  distinction's  sake 

A  picket,  or  station-staff  being  set  up  in  the  direction  of  the  line 
to  be  measured,  if  there  do  not  appear  some  marks  naturally  in  that 
direction,  they  measure  straight  towards  it,  the  leader  fixing  down 
an  arrow  at  the  end  of  every  chain,  which  the  follower  always  takes 
up,  as  he  comes  at  it,  till  all  the  ten  arrows  are  used.  They  are 
then  all  returned  to  the  leader,  to  use  over  again.  And  thus  the 
arrows  are  changed  from  the  one  to  the  other  at  every  10  chains' 
length,  till  the  whole  line  is  finished ;  then  the  number  of  changes 
of  the  arrows  shows  the  number  of  tens,  to  which  the  follower  adds 
the  arrows  he  holds  in  his  hand,  and  the  number  of  links  of  another 
chain  over  to  the  mark  or  end  of  the  line.  So,  if  there  have  been 
3  changes  of  the  arrows,  and  the  follower  hold  6  arrows,  and  the 
end  of  the  line  cut  off  45  links  more,  the  whole  length  of  the  line  is 
set  down  in  links  thus,  3645. 

In  all  these  measures  horizontal  distances  are  required,  and  they 
are  obtained,  at  least  very  nearly,  by  holding  the  chain  in  a  horizon- 
tal position,  both  on  ascending  and  descending  ground.  If  the  de- 
clivity is  too  great  to  admit  of  measuring  a  whole  chain  at  a  time, 
take  a  part  of  it,  and  in  all  cases  the  proper  position  of  the  elevated 
extremity  should  be  determined  by  a  plumb  line.  The  reason  of 
these  operations  is  obvious  by  the 
adjoining  figure ;  we  require  the 
line  AJB,  and  not  the  line  along 
the  ground  as  AC. 

AJB=ab+cd-\-fC. 

It  is  not  only  necessary  to 
measure  lines  but  we  must  also 
know  their  direction  or  the  angles 
which  they  make  with  the  meridian. 

This  is  commonly  determined  by  means  of  the 
SURVEYOR'S    COMPASS. 

The  surveyor's  compass  consists  of  a  horizontal  circle  to  which 
are  attached  sight-vanes  and  a  magnetic  needle  delicately  balanced 
on  its  centre. 

When  the  compass  is  set,  that  is,  standing  in  a  free  horizontal 


MENSURATION  OF  LANDS. 


83 


position  and  the  needle  free  to  move  on  the  center,  the  needle  will 
keep  the  magnetic  meridian,  and  the  circular  plate  may  be  turned 
under  it  to  bring  the  sight-vanes  to  any  line ;  —  the  needle  will  then 
point  out  the  degree  of  inclination  which  the  line  makes  with  the 
meridian. 

It  is  important  that  this  part  of  the  subject  be  most  clearly  under- 
stood by  the  learner,  we  therefore  give  the  following  minute  illus- 
tration of  it. 

Let  the  reader  now  face  the  north,  with  the  book  open  before  him,  his 
right  hand  is  then  toward  the  east,  and  his  left  hand  toward  the 
west. 

The  following  fig- 
ure represents  the 
compass  set  to  the 
magnetic  meridian, 
that  is  the  sight-vanes 
Vv  and  the  needle 
lie  in  the  same  direc- 
tion. The  degrees  on 
the  plate  are  num- 
bered both  ways  from 
N and  S  to  #  and  W. 

At  the  first  view 
of  this  subject,  it  has 
surprised  many  to 
find  TFfor  west  on  the 
right  hand  toward  the 
east,  and  E  in  the 
direction  toward  the  west, 
next  figure. 

Suppose  we  wished  to  find  the  direction  of  a  line  from  the  center 
of  the  compass  to  the  object  B.  We  set  the  compass,  that  is 
place  it  horizontal  on  its  staff  or  tripod,  the  needle  will  take  the 
same  direction  as  in  the  first  figure,  parallel  to  the  margin  of  the 
paper. 

The  sight-vanes  Vv  are  turned  toward  the  object  B  which  turns 
the  whole  plate,  but  the  needle  retains  its  position. 


The  reason  of  this  is  explained  by  the 


84  SURVEYING. 

We  now  read  the 
degree  pointed  out 
by  the  north  end  of 
the  needle,  and  we 
find  it  to  be  about 
50°  on  the  arc 
between  N  and  Et 
showing  thai  the 
course  or  direction 
from  the  center  of  the 
com^pass  to  B  is  North 
about  50°  toward  the 
East  —  a  result  obvi- 
ously true. 

Turning  the  sight- 
vanes  toward  the 
north  west  will  bring 
the  arc  between  JTand  W  to  the  north  point  of  the  needle.  For 
example,  if  it  were  required  to  run  a  line  North  31°  West,  from  a 
certain  point,  all  we  have  to  do  is  to  set  the  compass  over  that  point, 
level  the  plate,  see  that  the  needle  is  free  to  move  on  its  pivot,  and 
so  turn  the  plate  that  the  north  end  of  the  needle  will  settle  at  31° 
between  ^V  and  W,  the  range  of  the  sight-vanes  will  then  show  the 
required  line.  Proceed  hi  the  same  manner  to  find  any  other  line. 

Care  should  be  taken  that  no  iron  or  steel  comes  near  the  compass 
while  operating  with  it.  To  insure  a  correct  position  of  the  needle 
is  the  principal  difficulty,  but  if  it  settles  with  a  free  motion,  descri- 
bing nearly  equal  arcs,  slowly  decreasing  on  each  side  of  a  given 
point  and  finally  rests  at  that  point,  it  operates  well,  and  may  be 
relied  upon. 

In  whatever  direction  we  run,  the  north  point  of  the  needle  should 
always  lie  on  some  part  of  the  north  side  of  the  plate,  that  is,  nearer 
to  .tYthan  to  S  and  this  can  always  be  except  when  we  run  due  east 
or  west  per  compass. 

Lines  should  be  tested  by  taking  back  sightx  or  reverse  bearings, 
which  will  be  exactly  in  the  opposite  point  of  the  compass,  in  case 
there  is  no  local  attraction  to  disturb  the  needle.  If  the  line  just 


MENSURATION  OF  LANDS.  85 

run  over  does  not  correspond  to  the  exact  opposite  point  of  the  com- 
pass, it  shows  carelessness  in  running  or  some  local  attraction  of  the 
needle.  We  shall  show  how  to  overcome  this  last  difficulty  fur- 
ther on. 

Compasses  are  usually  marked  to  half  degrees,  some  of  them  are 
subdivided  to  one  fourth  degrees,  but  by  the  aid  of  a  vernier  scale 
we  can  tlieoretically  read  the  arc  to  one  minute  of  a  degree. 

DESCRIPTION     OF     THE    VERNIER. 

The  vernier  to  a  compass  is  on  the  outer  edge  of  the  graduated 
limb.  It  is  a  slip  of  metal  made  to  fit  the  graduated  limb  of  an  instru- 
ment, and  the  equal  divisions  upon  it  are  so  made  that  n  divisions  on 
the  vernier  will  cover  nil  divisions  on  the  limb. 

The  vernier  of  the  compass  is  on  the  outside  of  the  dial  plate  and 
it  is  firmly  attached  to  the  bar  that  holds  the  sight-vanes.  The 
dial  plate  can  be  moved  to  and  fro  along  it  by  means  of  a  screw. 

The  vernier  is  used  when  the  needle  points  between  two  divisions 
on  the  limb  :  the  dial  plate  is  then  gently  moved  by  the  screw  until 
the  needle  points  exactly  to  the  preceding  division  on  the  limb,  this 
being  done,  that  division  on  the  vernier  which  makes  a  right-line, 
that  is,  coincides  with  a  division  on  the  arc  is  the  number  of  small 
divisions  (minutes)  to  be  added  to  the  division  on  the  limb  now 
pointed  out  by  the  needle. 

Practical  and  experienced  men,  never  use  the  vernier  of  the  compass, 
because  they  can  read  the  compass  without  it  to  greater  accuracy 
than  they  can  really  run  a  line. 

But  as  the  vernier  scale  is  of  the  greatest  importance  attached  to 
several  other  instruments,  which  will  be  referred  to  in  this  work,  we 
now  make  an  effort  to  give  the  learner  a  clear  comprehension  of  it. 

Let  AB  represent  a 
portion  of  an  arc  and  ED 
the  vernier  which  is  con- 
ceived to  be  attached  to 
an  index  bar  and  made 
to  revolve  with  it. 

In  case  0  on  the  vernier  makes  a  right  line  with  10°  on  the  arc 
as  is  represented  in  the  figure,  then  the  index  marks  10°.  But  if 
0  on  the  vernier  is  a  little  beyond  10°  we  then  look  along  the  vernier 


86  SURVEYING. 

scale  to  see  what  division  of  it  makes  a  right  line  with  some  division 
on  the  limb,  suppose  the  division  8  on  the  vernier  coincided  with  a 
division  on  the  limb  then  the  index  would  mark  10°  8'. 

To  understand  the  philosophy  of  this :  Let  x  represent  the  value 
of  a  division  on  the  vernier  and  n  the  number  of  them  which  cover 
(n — 1)  divisions  on  the  limb,  then 

nx=n — 1 


&c.  <fcc.  to  n,  number,  from  which  it  appears  that  one  division  on 
the  vernier  beyond  a  division  on  the  limb  corresponds  with  the  nth 
part  of  the  unit  of  graduation,  two  divisions  of  the  vernier  above 
two  divisions  on  the  limb,  correspond  with  2rcth  division  of  th<?  unit 
of  graduation,  &c. 

In  our  figure  n=30,  the  graduation  of  the  limb  is  to  half  degrees 
or  30  minutes,  and  this  vernier  measures  minutes.  Verniers  on  many 
instruments  measure  as  small  as  ten  seconds  of  arc  and  on 
very  large  instruments  as  low  as  four  seconds. 


MENSURATION    OF    LANDS.  87 


CHAPTER    II. 

Having  shown  in  the  preceding  chapter  how  to  use  a  compass  — 
to  run  lines,  and  to  measure  them  ;  the  next  step  is  to  keep  a 
proper  record  of  all  the  lines  run,  and  compute  the  areas  they  enclose. 

A  line  traced  on  the  ground,  is  called  a  course,  the  angle  that  it 
makes,  with  the  meridian  passing  through  the  point  of  beginning,  is 
called  its  bearing. 

A.  course  written  JV  42°  E,  indicates  that  the  line  runs  between 
the  north  and  the  east,  and  makes  an  angle  of  42°  with  the  meridian ; 
when  between  the  north  and  west,  we  write  N.  W.,  putting  the 
number  of  degrees  and  minutes  between. 

Lines  from  the  south  point,  are  also  written  S.  E.  and  S.  W. ; 
that  is,  bearings  are  reckoned  from  the  north  and  south  points,  east 
and  west,  as  the  case  may  be. 

Hence,  to  make  a  record  of  a  survey,  all  we  have  to  do  is  to  write 
the  bearing  and  distance  of  each  course,  and  if  the  last  side  runs  to 
the  point  of  beginning,  it  is  a  complete  survey  ;  otherwise  it  is  not. 

Of  course,  no  area  can  be  attached  to  any  un-enclosed  space.  To 
complete  a  partial  survey,  to  enclose  a  space,  to  find  an  area,  or  to 
test  the  accuracy  of  a  complete  survey  ;  the  most  satisfactory  method 
of  investigation,  is  that  known  as 

LATITUDE     AND      DEPARTURE. 

Latitude  is  the  distance  of  the  end  of  a  line  north  or  south  of  its 
beginning,  measured  on  a  meridian,  and  it  is  called  either  northing 
or  southing,  according  as  the  line  runs  north  or  south.  Departure 
b  the  distance  of  the  end  of  a  line  east  or  west  of  its  beginning, 
measured  perpendicular  to  a  meridian,  and  it  is  called  easting  or 
westing,  according  as  the  line  runs  east  or  west. 

For  example,  suppose  that  we  have  the  following  bearings  and 
distances,  which  enclose  a  space  represented  by  AJBCD. 

Bearings.  Distances. 

AJB  N.23°E.  -        -          17 

BC  N.  83°  #  -        -      11 

CD  S.\4°E.  -  23 

DA  N.  77°  W.  -        -        -      23.66 


88 


SURVEYING. 


Let  NS  represent  the  meridian 
running  through  A,  the  most  western 
point  of  the  field  ;  make  the  angle 
JV^J9=23°,  and  AB=17  ;  then  Ab 
is  the  latitude,  and  Bb  is  the  depart- 
ure,  corresponding  to  the  course 
AB.  By  means  of  the  right  angled 
triangle  A  Bb,  having  the  hypotenuse 
AB,  and  the  angles,  we  can  com- 
pute Ab,  and  Bb,  15.65,  and  6.64, 
or  we  can  turn  to  the  traverse  table, 
and  under  23°  and  opposite  17,  we 
shall  find  the  value  of  these  lines  at 
once  ;  and  this  is  the  utility  of  having  the  traverse  table. 

In  the  same  manner  we  find  Bm  and  m  C,  the  latitude  and  depart- 
ure corresponding  to  the  bearing  and  distance  of  the  line  BO.  We 
find  Bm=1.34,  and  m (7=  10.92. 

Thus  we  go  round  the  field,  taking  the  latitude  and  departure  of 
each  side,  and  arrange  the  whole  in  a  table  as  follows  : 


Bearings. 

Dist. 

N. 

s. 

E. 

w. 

AB        N.  23°  E- 
EG        N.  83°^- 
CD         S.  14°^. 

DA        N.  77°  W. 

17 
11 

23 
23,66 

15,65 
1,34 

5,33 

22,32 

6,64 
10,92 
5,56 

23,05 
23,05 

22,3222,32 

23,12 

When  the  several  operations  are  performed  with  perfect  accuracy, 
the  sum  of  the  northings  will  be  equal  to  that  of  the  southings,  and 
the  sum  of  the  eastings  to  that  of  the  westings.  This  necessarily 
follows  from  the  circumstance  of  the  surveyor's  returning  to  the 
place  from  which  he  set  out ;  and  it  affords  a  means  of  judging  of 
the  correctness  of  the  work.  But  it  is  not  to  be  expected  that  the 
measurements  and  calculations  in  ordinary  surveying  will  strictly 
bear  this  test.  If  there  is  only  a  small  difference,  as  hi  the  above 
example,  between  the  northings  and  southings,  or  between  the  east- 
ings and  westings,  it  may  be  imputed  to  slight  imperfections  in  the 
measurements. 

Here  the  northings  and  southings  agree,  but  the  eastings  are  a 
little  greater  than  the  westings ;  we  will  therefore  decrease  the 


MENSURATION    OF    LANDS.  89 

eastings  by  half  the  error,  and  increase  the  westings  by  the  same 
amount ;  the  sums  will  then  agree. 

We  do  this  without  any  formal  statement,  but  the  operation  is 
strictly  that  of  proportion  ;  the  greater  the  line  the  greater  the  cor- 
rection to  be  applied. 

When  the  errors  are  considerable,  a  re-survey  should  be  made, 
and  if  the  errors  are  still  great,  and  in  the  same  direction,  there  is 
reason  to  suspect  that  some  local  attraction  disturbs  the  free  action 
of  the  needle  ;  and  then,  if  the  importance  demands  it,  a  survey 
can  be  taken  witkoiti  the  compass,  by  methods  we  shall  explain 
in  some  following  chapter. 

We  shall  make  use  of  this  example  and  this  figure  to  illustrate 
the 

TAKING  OF  ANGLES  BY  THE  COMPASS. 

For  this,  and  for  several  other  operations  in  practical  mathematics, 
the  learner  must  not  expect  a  written  rule  :  original  principles  are 
far  more  simple  and  reliable. 

We  now  require  the  angle  A B  C ;  conceive  the  AE  to  be  pro- 
duced, then  the  angle  between  EC  and  the  produced  part,  is  83° 
less  23°,  or  60°.  Now  60°  taken  from  180°  gives  120°  for  the 
angle  ABC. 

Again  the  line  B  C  makes  an  angle  with  the  meridian  toward  the 
north  of  83°,  therefore  toward  the  south  it  must  be  97°  on  the 
east  side  of  it.  The  line  BA  makes  an  angle  of  23°  with  the  meri- 
dian on  the  west  side  of  it;  therefore,  the  angle  ABC—  97+23 
=  120,  the  same  as  before. 

To  find  the  angle  BCD,  we  add  83°  and  14°.     Why  ? 

To  find  the  angle  CD  A,  we  subtract  14°  from  77°.    Why  ? 
o  find  the  angle  DAB,  we  add  23°  to  77°,  &c. 
:'he  sum  of  these  4  angles  must  equal  4  right  angles. 

Suppose  now  that  the  surveyor  runs  the  lines  AB,  BC,  CD,  and 

ien  wishes 

TO     CLOSE     THE     SURVEY. 

To  close  a  survey  is  to  run  the  last  side  so  as  to  strike  the  first 
oint,  when  we  are  not  able  to  see  it. 
To  accomplish  this,  we  sum  up  the  latitude  and  departure  as  far 


90  SURVEYING 

as  the  point  D,  the  result  will  show  Dd  and  dA.  Having  then  two 
sides  of  the  right  angled  triangle,  the  angle  dAD  will  be  the  bear- 
big  for  dAD  =  nDA,  because  nD  and  NS  are  parallel.  The  side 
DA  can  also  be  computed,  but  it  should  be  measured  also,  as  a  test 
to  the  accuracy  of  the  whole  survey.  If,  on  running  DA,  accord- 
ing to  computation,  we  actually  strike  the  point  A,  or  very  near  to 
it,  and  there  is  little  or  no  difference  between  actual  measure  and 
computation,  then  we  may  be  sure  that  all  the  sides  have  been  run 
correctly ;  but  if  on  running  DA,  we  do  not  strike  A,  or  the  dis- 
tance does  not  correspond  to  computation,  we  may  be  sure  of  errors 
somewhere  —  either  in  want  of  skill  or  care  in  the  operation,  or  the 
action  of  the  compass  has  not  been  uniform  at  all  the  angular  points. 
In  case  of  material  errors  a  re-survey  should  be  made. 

In  case  that  we  have  no  means  of  making  computations  in  the 
field,  we  may  take  a  course  as  near  the  true  one  as  our  judgment 
will  permit,  and  run  it.  This  line  must  bring  us  near  the  point  of 
beginning,  if  it  does  not  strike  it,  and  when  we  get  opposite  to  that 
point  we  must  measure  to  it  at  right  angles  from  the  line  run  ; 
then  we  shall  have  data  to  correct  our  course.  Running  a  line  thus 
by  guess  work,  is  called  running  a  random  line,  from  which  the  true 
line  can  be  found  as  follows  : 

Suppose  that  when  we  arrive  at  D,  we  judge  the  course  to  the 
first  point  to  be  N.  75°  IF".,  and  run  that  course,  and  after  measuring 
28.65  chains  we  find  that  we  are  passing  the  first  point,  which  is  83 
links  in  perpendicular  distance  toward  the  south;  what  course 
should  have  been  taken  ? 

By  the  following  investigation  we  draw  out  a  rule  that 
will  apply  to  all  such  cases. 

Let  AB  represent  a  true  course,  AD  a  random  Hner 
and  DJB  its  amount  of  deviation. 

Also  let  DAE  equal  one  degree,  and  take  Ad—1, 
then  the  deviation  at  d  will  be  the  natural  sine  of  one 
degree,  and  may  be  taken  from  the  table  of  natural  sines 
(which  is  .01745). 

By  proportional  triangle  we  have 

I  :.01745:  \AD\DE\ 

Whence  D£=Q.Q1745(AD). 

Now  J9^is  contained  in  DJB  as  often  as  1°  is  contained  in  the 


MENSURATION  OF  LANDS.  91 

number  of  degrees  in  the  angle  DAB.    Let  x  represent  the  num- 
ber of  degrees  hi  DAB,  then 

x_^       (DB) 

I     .On45(AD)' 


That  is,      x--9    because 
(AD) 

Hence,  to  correct  a  course,  we  have  the  following 

RULE.  —  Multiply  the  deviation  by  57.3,  and  divide  that  product  by 
the  distance,  and  the  quotient  will  be  the  number  of  degrees  and  parts 
of  a  degree  to  add  to,  or  subtract  from,  the  random  course. 

This  rule,  applied  to  the  present  example,  gives 
.83(57.3)_g0 
23.65 

Hence,  the  true  course  is  75°+2°=770. 

Had  the  deviation  of  the  random  line  been  toward  the  south,  we 
should  have  subtracted  the  correction  ;  but  for  this  the  operator 
must  rely  on  his  judgment. 

We  now  come  to  the 

COMPUTATION     OF     AREAS. 

By  inspecting  the  figure  we  per- 
ceive that  cCDd  is  a  trapezoid, 
from  which,  if  we  subtract  the  trian- 
gles ADd,  ABb,  and  the  trapezoid 
IBCc,  the  area  of  the  field  ABGD 
will  be  left. 

OBSERVATION.  —  To  preserve  uniformity 
of  expression,  and  clearness  and  brevity 
in  forming  a  rule,  we  shall  call  triangles, 
trapezoids,  while  discussing  this  subject. 
A  trapezoid  becomes  a  triangle,  when  its 
smallest  parallel  side  is  so  small  as  to  call 
it  zero  —  conversely,  then,  a  triangle  is  a 
trapezoid,  whose  smallest  parallel  side  is  zero. 

We  observe  that  C  is  the  most  northern  point  of  the  field,  and  D 
is  the  most  southern.  In  traversing  from  C  to  D,  from  the  north  to 
the  south,  we  pass  along  the  oblique  sides  of  trapezoids  that  we 
shall  call  south  areas,  and  hi  traversing  from  D  to  A,  B,  and  C, 


*W  SURVEYING. 

from  the  south  to  the  north,  we  pass  along  the  oblique  sides  of 
trapezoids,  which  we  shall  call  north  areas.  Now  it  is  obvious  that 
if  we  subtract  the  sum  of  the  north  areas  from  the  sum  of  the  south 
areas,  we  shall  have  a  remainder  equal  to  the  area  of  the  field. 

We  now  require  a  systematic  method  of  finding  these  areas,  or 
the  area  of  these  several  trapezoids.  In  the  first  place,  we  must 
have  latitudes  and  meridian  distances. 

Latitude  and  departure  have  already  been  defined  and  explained. 

MERIDIAN     DISTANCES. 

Meridian  distances  are  the  distances  of  the  angular  points  of  the 
field  from  the  meridian  which  runs  through  the  most  westerly  point 
of  the  field ;  thus,  bJB,  cC,  dD,  are  meridian  distances. 

Double  meridian  distances  are  the  bases  of  triangles,  or  the  sum  of 
the  parallel  sides  of  the  trapezoids. 

Thus  IB  is  the  double  meridian  distance  of  the  side  AB,  or  it  is 
the  double  meridian  distance  of  the  middle  point  of  the  line  AB. 
bB  +  cC  is  the  double  meridian  distance  of  the  line  BC,  or  double 
the  meridian  distance  of  the  middle  point  of  BO.  This  double 
meridian  distance  (bB-}-cC)  multiplied  by  bC,  or  Bm,  the  latitude 
corresponding  to  BC,  will  give  the  double  area  of  the  trapezoid 
bB,  Cc. 

We  are  now  prepared  to  give  the  following  summary  or  rule  for 
finding  the  area  of  any  field  bounded  by  any  number  of  right  lines : 

RULE. —  1.  Prepare  a  table  headed  as  in  the  example,  namely  : 
Bearings,  Distance,  North,  South,  East,  West,  Meridian  distance, 
Double  meridian  distance,  North  areas,  South  areas. 

2.  Begin  at  the  most  western  point  of  the  field,  and  conceive  a  meri- 
dian to  pass  through  that  point. 

Find,  by  the  traverse  table  or  by  trigonometry,  the  northings,  southings, 
eastings,  and  westings  of  the  several  sides  of  the  field,  and  set  them  in 
the  table  opposite  their  respective  stations,  under  their  proper  letters  aV., 
S.,  E.,  or  W. 

3.  For  the  first  meridian  distance  take  the  departure  of  the  first  line  ; 
for  the  second,  take  the  first   meridian  distance  and  add  to  it   th* 
departure  of  ike  second  line,  if  the  departure  is  east,  or  subtract  if 
west,  <&c. 


MENSURATION    OF    LANDS. 


93 


4.  Add  each  two  adjacent  meridian  distances,  and  set  their  sum 
opposite  the  last  of  the  two  in  the  column  of  double  meridian  distances. 

5.  Multiplg  each  double  meridian  distance  by  the  latitude  to  which  it 
is  opposite,  and  set  the  product  in  the  column  of  N.  areas,  if  the  latitude 
is  north,  and  in  that  of  S.  areas,  if  the  latitude  is  south. 

6.  Subtract  the  sum  of  the  N.  areas  from  that  of  the  S.  areas,  and 
take  half  the  remainder,  which  will  be  the  area  of  the  field  in  square 
chains.     Dividing  this   by  10  gives  the  acres ;   and  the  roods  and 
rods  are  found  by  multiplying  the  decimal  parts  by  4  and  by  40. 


AB 
EC 
CD 
DA 

Bearings. 
N  23°  E 

N  83°  E 
S  14°  E 

N11°W 

Dis. 

N. 

S. 

E. 

w. 

M.   D. 

D.  M.D. 

N.  areas.  S.  areas 

17 

11 

23 

23.66 

15.65 
1.34 

5.33 

22.32 

6.64 

(6.63) 
10.92 
(10.90) 
5.56 

(5.55) 

23.05 

(23.08) 

6.63 
17.53 

23.08 
0.00 

Div 

6.63 
24.16 
40.61 
23.08 

103.76 
32.37 

123.02 

906.42 

23.08 

259.15 

Diff., 

Half, 
idingby  10, 

906.42 
25<U5 

K47.27 

323.63 
32.363 

Hence  the  field  contains  323  square  chains  and  63  hundredths,  or 
thirty  two  acres  and  a  little  more  than  36  hundredths  of  an  acre. 

The  numbers  in  parentheses,  as  (6.63),  and  all  others  in  parenthe- 
ses are  the  numbers  corrected  to  make  the  eastings  and  westings 
agree,  —  the  numbers  above  them  are  taken  from  the  table. 

Before  we  give  any  more  examples,  it  is  proper  to  give  some 
examples  to  show  the  practical  utility  of  the 

TRAVERSE    TABLE. 

This  table  is  computed  to  every  half  degree,  but  if  a  course  is 
between  two  courses  in  the  table,  the  operator  can  use  his  judgment 
and  take  out  the  proper  intermediate  numbers. 

Those  who  are  not  satisfied  with  this  method,  can  use  the  table 
of  natural  sines  and  cosines,  as  we  shall  subsequently  explain. 

The  distances  are  consecutive  to  30,  then  35,  40,  <fec.,  to  100. 
But  a  little  thought  hi  the  operator  will  enable  him  to  use  the  table 
for  any  distance  whatever. 


94  SURVEYING. 

The  following  examples  will  illustrate. 

1.  A  course  is  N.  22°  30'  E.  distance  62.43  ;  what  is  the  corres- 
ponding latitude  and  departure,  as  found  in  the  table  ? 

We  shall  regard  the  distance  as  6243  links,  and  separate  it  into 
parts. 

Lat. 

Thus:  6000  5543 

240  221.7 

3  2.77 

6243  5767.47  2388.95 

If  we  now  return  to  chains  and  links,  the  latitude  is  57.67  and  the 
departure  23.89. 

We  entered  240  in  the  table,  as  24  chains,  and  took  the  numbers 
corresponding. 

2.  A  course  is  N.  48°  30'  W.y  distance  187.61  ;  what  is  the  corres- 
ponding latitude  and  departure  ? 

Dis.  Lat.  Dep. 

180.00  119.30  134.80 

7.60  5.036  5.692 
1                               066  075 

187.61  124.3426  140.4995 

If  we  now  take  the  distance  as  1 87  chains  and  6 1  links,  the  Lat. 
is  124  ch.  34  links,  and  the  Dep.  is  140  ch.  and  50  links. 

3.  A  course  is  &  81°  W.,  distance  76.87  ;  what  is  the  corres- 
ponding latitude  and  departure  ? 

Dis.  Lat.  Dep. 

75.00  1173. 

1.80  28.2 
7  1.10 

76.87  1202.3  7592.71 

If  76  ch.  87  lin.,  Lat.  12  ch.  2  lin.,  Dep.  75  ch.  93  links. 

Thus  we  can  find  the  latitude  and  departure  for  any  distance  cor- 
responding to  any  degree  and  half  degree. 

We  can  find  it  to  any  degree  and  minute  of  a  degree  by  the  table 
of  natural  shies  and  cosines. 

The  common  tables  containing  natural  cosines  and  sines  are  nothing 
more  than  latitude  and  departure  corresponding  to  unity  of  distance. 


MENSURATION   OF   LANDS.  95 

Therefore,  a  double  distance  will  correspond  to  a  double  distance 
in  latitude  and  departure,  a  treble  distance  will  give  a  treble  amount 
of  latitude  and  departure,  and  so  on  in  proportion.  Lat.  «=  Nat. 
cosine.  Dep.  =  Nat.  sine. 

Hence  :  The  natural  cosine  of  any  course  taken  as  a  decimal,  mul- 
tiplied by  any  distance,  ivill  give  the  latitude  corresponding  to  that 
course  and  distance.  Also,  the  natural  sine  taken  as  a  decimal,  multi- 
plied by  a  given  distance,  will  give  the  departure  corresponding  to  that 
course  and  distance. 

N.  B.  Nat.  sines  and  cosines  are  found  in  table  II.,  pages  21-65 
of  tables.  For  common  purposes,  four  places  of  decimals  are  suffi- 
cient. 

1.  The  bearing  of  a  certain  line  is  N.  35°  18'  E.  ;  distance   12 
chains ;   what  is  the  corresponding  latitude  and  departure  ? 

Angle  35°  18'       N.  cos.  .81614  N.  sin.         .57786 

Dis.  (multiplier)  12 

Diff.  Lat=  9.79368  Dep. 

2.  A  certain  line  runs  S.  4°  50'  E. ;  distance  74.40  ;  what  is  the 
corresponding  latitude  and  departure  ? 

Angle  4°  50'         N.  cos.  .9964  N.  sin.         .0842 

Distance  74.4  74.4 

1*9856  ~3368 

39856  3368 

69748  5894 


Lat.  74.13216  Dep.         6.26448 

3.  A  line  makes  an  angle  with  the  meridian  of  75°  41',  at  a  dis- 
tance of  89.75  chains ;  what  is  the  latitude  and  departure  ? 

75°  47'         cos.  .2456  sin.  .9694 

Distance  89.75  89.75 

Prod.  Diff.  Lat.  22.042  Dep.  87.001 

4.  A  line  bearing  N.  7°  40'  W.  ;  distance  31.20  chains  ;  required 
the  difference  of  latitude  and  departure. 

7°  40'  cos.  .98106  sin.  .13341 

Multiplier  31.2  31.2 

.  Lat.  "3O92  Dep.  Tl6~" 


96  SURVEYING. 

5.  A  line  running  S.  80°  10'  E.     distance  35.25  chains  ;  what  is 
the  difference  of  latitude  and  departure  ? 

80°  10'         cos.  .17078  sin.  .9853 

Multiplier  35.25  35.25 

Diff.  Lat.  6.02  Dep.  34.72 

In  the  last  three  examples,  we  have  given  only  the  results  of  the 

multiplications  to  two  places  of  decimals  ;  that  is,  to  the  nearest 

link,  which   is   a   degree   of   accuracy  sufficient  for   all  practical 

purposes. 

We  are  now  prepared  to  estimate  the  areas  of  the  following  general 

surveys,  given  as 

EXAMPLES. 
1.  In  May,  1845,  the  following  measures  of  a  field  were  taken. 

Beginning  at  the  western-most  point  of  the  field  ;  thence  N.  20°  30' 

E.  5  chains  83  links;   thence  S.  79°  45'  E.   10  chains    15  links; 

thence  S.  27°  30'    W.  9  chains  45  links;  thence  ^Y.  63°  15'    W. 

8  chains  28  links  ;  thence  N.  15°  30'  W.  1  chain  and  4  links,  to  the 

place  of  beginning ;  required  the  area. 

It  is  not  absolutely  necessary 
to  make  a  plot  or  figure  of  the 
field,  but  for  the  sake  of  per- 
spicuity, it  is  best  to  do  so ; 
yet  no  reliance  is  placed  on  the 
accuracy  of  the  constructed 
figure. 

We  perceive  by  the  figure, 
that  there  are  two  south  areas, 
bBCc,  and  cCDd  ;  and  three 
north  areas,  eEdD,  AeE,  and 
AM. 

Let  the  reader  observe  that 
all  the  north  areas  are  on  the 
outside  of  the  field. 


MENSURATION    OF    LANDS. 


AB 
BC 
CD 
DE 
EA 

Bearings. 

S.79°45'.E. 
S.27°30'W. 
AT.63°15'W. 
N  1  5030'  W. 

Dis. 

io!is 

9.45 

8.28 
1.04 

N. 

S. 

E. 

w. 

M.  D. 

D.M.D. 

.V.  areas. 

S.  areas. 

5.46 

3.73 
1.00 

10.19 

1.81 

8.38 

2.04 
9.99 

4.36 
7.39 
0.28 

2.04 
12.03 
7.67 
0.28 
0.00 

2.04 
14.07 
19.70 
7.95 
0.28 

11.1384 

29.6535 
0.2800 

25.4667 
165.0860 

10.19 

12.03 

12.03 
Area  in  acres, 

41.0719  190.5527 
41.0719 
2)149.4808 
10)74.7404 
7372" 

The  operator  can  rely  on  the 
rule  used  in  the  last  operation, 
whatever  be  the  number  of  sides, 
or  whatever  be  the  shape  of  the 
figure,  provided  that  the  lines  are 
right  lines,  from  one  angular  point 
to  another. 

In  case  of  re-entering  angles, 
like  HI  A,  represented  in  the  ad- 
joining figure,  a  portion  of  the 
figure  Air,  is  reckoned  twice.  But 
this  is  corrected  by  the  subtractive 
space  fflih,  which  includes  not 
only  the  exterior  portion  hBIA, 
but  also  the  whole  additive  trian- 
gle All,  belonging  to  the  last  side 
of  the  figure  I  A. 

In  the  following  example  are  several  such  re-entering  angles. 

2.  Find  the  area  of  a  lot  of  land,  of  which  the  following  are  the 
field  notes. 

Beginning  at  the  south  west  corner,  the  ancient  land  mark ;  thence 

1.  N.  27°  15'  E.      distance      9.42  chains. 

2.  S.  80°  00'  E. 


3.  S.  69°  00'  E. 

4.  S.  15°  45' W. 

5.  N.  66°  45'  W. 

6.  S.  31°00'TF". 

7.  N.  70°  45'  W. 

8.  S.  41°  45'  W. 

9.  JV.  63°  00' IF. 


1.15 
12.73 
5.00 
1.05 
2.90 
8.92 
2.08 
4.18 


SURVEYING. 


We  can  contract  the  operation  in  reference  to  the  space  it  will 
occupy,  by  putting  the  difference  of  latitude  in  one  column,  and  all 
the  departures  in  another  column. 

Marking  all  the  northings  by  the  sign  -f-,  and  all  the  southings  by 
the  sign  — .  Also,  all  the  eastings  by  the  sign  -f-  >  and  all  the 
westings  by  the  sign  — . 

This  being  understood,  the  work  will  appear  as  follows  : 


Bearings. 

Dis.  I 

Lat.          Dep. 

M.  D.  |D.M.D.     N.  areas       S.  areas 

1  N.  27o  15'  E, 

9.42 

4-8.37 

4-  4.31 

4.31 

4.31    36.0747 

2  S.  800  00'  E. 

1.15 

—0.20 

4-  1.13 

5.44 

9.75 

1.9500 

3  S.  690  00'  E. 

12.73 

—4.57 

4-11.89 

17.33 

22.77 

104.0589 

4  S.  15C  45'  W. 

5.00 

—4.81 

—  1.36 

15.97 

33.30 

160.1730 

5  'N.  66°  45'  W. 

1.05 

4-0.41 

—  0.96 

15.01 

30.98 

12.7018 

6  S.  31°  00'  W. 

2.90 

—2.49 

—  1.49 

13.52 

28.53 

71.0397 

7  JV.  700  45'  HT 

8.92 

4-2.94 

—  8.42 

5.10 

18.62 

54.7428 

8  S.  410  45'  w. 

2.08 

—1.55 

—  1.38 

3.72 

8.82 

13.6710 

9  N.  630  00'  W. 

4.18 

4-1.90 

—  3.72 

0.00 

3.72 

7.0680 

Sum 

(MM) 

0.00 

110.5873 

350.8926 

110.5873 
2)240.30f 
10)120.1526 
12  acres,  and  a  small  fraction  over.  12.0152GJ 

3.  Having  the  following  field  notes,  it  is  required   to  find  the 
closing  side  and  the  area  of  the  field. 


Bearings.         j 

Dis.    I 

N. 

S.      | 

E. 

w. 

1  S.  75°         W. 
2  S.  20°  30'  W. 
3           W. 
4JV.330  30'.#. 
5JV.760        E. 
6         South 

13.70 
10.30 
16.20 
35.30 
16.00 
9.00 

29.51 
3.87 

3.54 
9.65 

9.00 

19.49 
15.52 

13.24 
3.60 
16.20 

33.38 
22.19 

11.19 

22.19 

35.01 
33.04 

33.04 

1.97 

This  result  shows,  that  if  we  commence  at  the  first  station,  and 
traverse  round  to  the  sixth,  we  shall  then  be  11.19  chains  to  the 
north  of  the  place  of  beginning,  and  1.97  chains  east  of  it.  This  is 
sufficient  data  to  compute  the  course  and  distance. 

To  compute  the  area,  however,  it  is  not  necessary  to  find  either  the 
course  or  the  distance. 

We  do  know,  however,  by  merely  inspecting  the  traverse  table. 


MENSURATION   OF   LANDS. 


that  the  course  to  the  place  of  beginning,  is  south  about  10°  20'  west, 
and  distance  near  12  chains. 

We  are  now  prepared  to  compute  the  area,  and  as  we  wish  to 
commence  at  the  western-most  point  of  the  field,  we  shall  begin  at 
the  4th  station,  calling  it  the  first :  thus, 


i            Bearings. 

Dis. 

Lat. 

Dep. 

M.D.  1 

19.49 
35.01 
35.01 
33.04 
19.80 
16.20 
0.  0 

D.M.D. 
19^49 
54.50 
70.02 
68.05 
42.84 
36.00 
16.20 

N.  area. 

S.  area. 

1  N.  330  30'  E. 
2  N.  760        £. 
3         South 
4         S.  W. 
5  S.  750         W. 
6  S.  200  30   w. 
7            W. 

35.30 
16.00 
9.00 

13.70 
10.30 
16.20 

4-29.51 
4-  3.87 
—  9.00 
—11.19 
—  3.54 
—  965 
0.  0 

4-19.49 
4-15.52 
0.  0 
—  1.97 
—13.24 
—  3.60 
—16.20 

575.1499 
210.9150 

630.1800 
761.4795 
151.6536 
347.4000 

i 

786.0649 

1890.7131 

This  result  shows  that  the  field  contains  55  acres,  and 
a  little  more  than  23  hundredths  of  an  acre. 


786.0649 
2)1104.6482 
10)552.3241 
55.23241 

4.  What  is  the  area  of  a  survey,  of  which  the  following  are  the 
field  notes 

Bearings. 

S.  46°  30'  E. 
S.  51°  45'  W. 

West. 

N.  56°        W. 
N.  33°15'K 
S.  74°  30'  E. 


Stations. 

1 

2 
3 
4 
5 
6 


Distances. 

80  rods. 
55.16 
85.00 
110.40 
75.20 
123.80 
Ans.  104.35  acres. 

5.  Required  the  contents  and  plot  of  a  piece  of  land,  of  which  the 
following  are  the  field  notes. 

Stations  Bearings. 

1  £.34°  W. 

2  S. 

3  S.  36i°JE, 

4  N.  59^. 

5  N.  25°  E. 

6  N.  16°  W. 

7  N.  65°  W. 


Distances. 

3.95  ch. 
4.60 
8.14 
3.72 
6.24 
3.50 
8.20 
Ans.  10.4.  Q&.  5P. 


100  SURVEYING. 

6.  Required  the  contents  and  plot  of  a  piece  of  land,  from  the 
following  field  notes. 

Stations.  Bearings.  Distances. 

1  S.  40°  W.  70  rods. 

2  N.  45°  W.  89 

3  N.  36°  E.  125 

4  N.  54 
6            S.  81°  E.                  186 

6  S.    8°  TT.  137 

7  TF.  130 

Ans.  207.4.  3R.  33P. 

7.  Given  the  following  bearings  and  distances  of  the  several  sides 
of  a  field,  namely, 

1.  N:  58°  E.  19  ch. 

2.  E.     6°  S.  20 

3.  £.  17°   W.  20 

4.  "FT.  20 

5.  N.  42°  35'  W.          15.10 
to  find  the  area. 

Ans.  54.9  acres. 

8.  Given  the  following  bearings  and  distances,  namely, 

1.  N.  45°  E.  40  ch. 

2.  £  30°   W.  25 

3.  /S.     5°  E.  36 

4.  TF  29.60 

5.  N.  20°  -#.  31 

to  find  the  corrected  difference  of  latitude  and  departure,  and  the 
area. 

N.  B. — In  this  last  example,  as  hi  most  others,  the  northings  and 
southings  will  not  exactly  balance  ;  nor  will  the  eastings  and  west- 
ings balance.  This  arises  from  inaccuracies  in  the  data.  In  such 
cases  (  if  the  errors  are  but  trifling  )  we  balance  off  the  errors. 

When  a  course  is  east  or  west,  as  the  4th  in  this  example,  some 
operators  have  expressed  doubts,  whether  any  correction  should  be 
applied  to  latitude  in  that  course. 

We  reply,  that  errors  do  really  exist,  and,  therefore,  we  cannot 


MENSURATION    OF    LANDS. 


101 


say  that  the  course  marked  west,  in  the  example,  was  really  west,  or 
not ;  the  probability  is,  that  the  course  was  not  exactly  due  west, 
and  it  is  therefore  proper  to  put  a  correction  in  the  latitude  column, 
as  shown  in  the  following  results : 

CORRECTED      LATITUDES      AND      DEPARTURES. 


» 

1. 

2. 
3. 
4. 
5. 

N. 

s. 

E. 

28.30 
3.16 
10.62 

w. 

28.30 

0.02 
29.15 

21.63 
35.84 

12.49 
29.59 

57.47 

57.47 

42,08 

42.08 

On  inspecting  these  latitudes  and  departures,  we  perceive  station 
6  is  the  most  westerly  point  of  the  field,  therefore  to  find  the  area, 
we  will  arrange  these  results  in  the  following  order  : 


Lat. 

Dep. 

M.  D. 

D.  M.  D. 

N.  areas. 

S.  areas. 

5    +29.15 
1    4-28.30 
2    —21.63 
3    —35.84 
4    -{-00.02 

+  10.62 
+28.30 
—12.49 
4-  3.16 
—29.59 

10.62 
38.92 
26.43 
29.59 
0.  0 

10.62 
49.54 
65.35 
56.02 
29.59 

309.5730 
1400.9820 

0.5918 

1403.5205 
2007.7568 

1711.1468 

3410.2773 
HI!.  1468 

2)1699.1305 

10)849.5652 

Ans.  areas,  84.956 

9.  What  is  the  area  of  a  survey  of  which  the  following  are  the 
field  notes. 

From  the  place  of  beginning,  JV7".  31°  30'  W.y  distance  10  chains  : 
thence  N.  62°  45'  E,,  9.25  chains  :  thence  5.  36°  E.,  7.60  chains  : 
thence  S.  45°  30'  W.t  10.40  chains,  to  the  place  of  beginning. 

Ans.  8TyT  acres. 

10.  Do  the  following  bearings  and  distances  enclose  a  space  ? 
If  not,  give  an  additional  bearing  and  distance  that  will,  then  deter- 
mine  the  area  so  enclosed. 


102 


SURVEYING. 


Stations.  Bearings.  Distances. 

1  S.  40°  30'  K  31.80  ch. 

2  N.  64°  00'  E.  2.08 

3  N.  29°  15'  E.  2.21 

4  N.  28°  45'  .£  35.35 

5  N.  57°  00'  TT.  21.10 

Ans.  These  bearings  and  distances  do  not  enclose  a  space.  A 
line  run  from  the  further  extremity  of  the  5th  to  the  first  station 
will  bear  south  46°  43'  W.9  distance  31.21  chains,  and  the  area  thus 
enclosed  will  contain  92.9  acres. 

11.  Do  the  following  bearings  and  distances  enclose  a  space  ?  If 
not,  determine  the  additional  line  that  will,  and  the  area  of  the  space 
so  enclosed. 

Stations.  Bearings.  Distances. 

1  S.  85°  00'  W.  46.4  rods. 

2  N.  53°  30'  W.  46.4     " 
*3            N.  36°  30'  E.  76.8     " 

4  N.  22°  00'  E.  56.0     " 

5  S.  76°  30'  E.  48.0     " 

Ans.  These  bearings  and  distances  do  not  enclose  a  space.  A 
Bne  run  from  the  last  station  to  the  first  would  bear  S.  \  3°  25'  W., 
distance  128.6  rods.  Area  54.86  acres  nearly. 

The  operation  for  the  area  is  as  follows : 

We  commence  at  station  3,  for  reasons  that  have  been  several 
times  explained.  Station  6  is  the  one  we  supplied. 


Sta.j    Lat. 

Dep. 

M.  D. 

45.65 
66.63 
113.30 
83.45 
37.30 
0.  0 

D.  M.  D. 

N.  area. 

S.  area. 

3 

4 
5 
*6 
1 

2 

-}-  61.73 
+  51.92 
—  11.21 
—125.18 
—  4.85 
+  27.59 

+45.65 
+20.98 
+46.67 
—29.85 
—46.15 
—37.30 

45.65 
112.28 
179.93 
196.75 
120.75 
37.30 

2817.9745 
5829.5776 

1029.1070 

2017.0153 
24629.1650 
585.6375 

19676.6590 
Si 

27231.8178 
9676.6590 
1755571588 

160)8777.5794(54.86. 


*  The  stations  marked  with  a  *  are  those  supplied. 


THE  MERIDIAN  LINE.  103 

12.  What  is  the  area  of  a  survey  of  which  the  following  are  the 
field  notes. 

Stations.  Bearings.  Distances. 

1  N.  75°  00'  E.  54.8  rods. 

2  N.  20°  30'  E.  41.2     " 

3  E.  64.8     « 

4  S.  33°  30'  W.  141.2     " 

5  8.  76°  00'  W.  64.0     " 

6  N.  36.0     " 

7  8.  84°  00'  IP!  46.4     " 

8  N.  53°  15'  W.  46.4      " 

9  JV.  36°  45'  J£  76.8      " 

10  N.  22°  30'  ^.  56.0  " 

11  S.  76°  45'  E.  48.0  " 

12  £  15°  00'  W.  43.4  " 

13  S.  16°  45'  PT.  40.5  " 

In  this  survey  4  is  the  most  easterly  and  9  the  most  westerly 
station.  The  area  is  equal  to  IIQA.  %R.  23J*.  It  may  vary  a 
little,  on  the  account  of  the  way  in  which  the  balancing  is  done. 


CHAPTER    III. 

ON    THE    MERIDIAN    LINE    AND    THE    VARIA- 
TION   OF    THE    COMPASS. 

THE  meridian  is  an  astronomical  line,  having  no  necessary  con- 
nection with  the  magnetic  needle.  Meridians  would  be  primary 
lines  to  which  we  would  refer  all  surveys,  if  there  were  no  such 
thing  as  magnetism,  or  a  magnetic  needle. 

It  is  only  a  coincidence,  that  the  magnetic  needle  settles  near 
the  meridian,  so  near,  that  for  a  long  time  it  was  considered  the 
meridian  itself,  but  accurate  observations  have  shown  that  the 
needle  does  not  point  rigorously  north  and  south,  but  has  a  variation 
which  is  not  the  same  at  all  times  in  the  same  place,  therefore  a 
line  run  by  the  compass  is  still  unknown  in  respect  to  the  meridian, 


104  SURVEYING. 

unless  we  know  the  variation  of  the  compass,  that  is,  the  declination 
of  the  needle. 

In  the  year  1657  the  needle  at  London  pointed  due  north,  since 
that  tune  its  variation  has  been  west,  previous  to  that  time  the  vari- 
ation was  east. 

In  the  Atlantic  ocean,  between  Europe  and  the  United  States,  the 
variation  is  from  12°  to  18°  west. 

The  needle  seems  to  point  to  the  region  of  greatest  cold,  which 
is  in  the  northern  part  of  America,  and  not  the  north  pole,  and  if 
this  be  true,  if  the  point  of  minimum  heat  changes  its  position,  there 
will  be  a  corresponding  change  in  the  direction  of  the  magnetic 
needle. 

The  needle  has  a  small  annual  and  also  a  diurnal  variation,  cor- 
responding to  the  temperature  of  the  different  seasons  of  the  year, 
and  of  the  different  times  of  the  day,  but  these  variations  are  too 
small  to  trouble  the  common  operations  of  surveying. 

Some  of  the  variations  of  the  compass  are  regular,  others  irregu- 
lar ;  some  amount  to  many  degrees  and  require  a  long  period  of 
time,  others  are  small  in  amount  and  require  but  short  intervals  to 
pass  through  all  their  changes. 

The  daily  variation  consists  of  an  oscillation  eastward  and  west- 
ward of  the  mean  position,  and  is  different  in  different  places. 
Generally  the  greatest  oscillation  eastward  is  between  six  and  nine 
in  the  morning,  and  westward  about  one  hi  the  afternoon,  gradually 
returning  toward  the  east  until  eight  P.  M.  At  night  it  is  sta- 
tionary. 

On  the  subject  of  magnetism  we  know  nothing,  beyond  facts 
drawn  from  observation,  but  there  is  no  doubt  that  the  earth  is  a 
great  magnet,  made  so  by  the  action  of  the  sun,  and  the  poles  of 
this  great  magnet  are  near  the  poles  of  the  equator.  Indeed,  all 
observations  made  correspond  to  this  hypothesis,  for  changes  of  the 
weather,  clouds,  and  storms,  all  have  an  influence  on  the  needle. 

These  facts  are  sufficient  to  convince  any  reader  that  to  survey 
correctly  we  must  know  the 

VARIATION     OP     THE     COMPASS. 

As  the  true  meridian  is  an  astronomical  line,  we  must  find  it  by 
astronomical  observations,  and  then  by  comparing  the  meridian  of 


VARIATION  OF  THE  COMPASS.  105 

the  compass  with  it,  we  shall  have  the  variation  of  the  compass. 

When  the  sun  is  on  the  equator,  it  rises  due  east,  and  sets 
directly  in  the  west.  Should  we  then  observe  the  direction  of  its 
center,  just  as  it  was  rising  or  setting,  at  the  time  it  had  no  declina- 
tion, and  trace  that  line  a  short  distance  on  the  ground,  we  should 
then  have  a  due  east  and  west  line. 

If  from  any  point  in  that  line  we  draw  another  line  at  right  angles, 
we  should  then  have  the  true  meridian. 

If  we  now  put  the  compass  on  this  meridian,  and  make  the  sight- 
vanes  range  with  it,  the  needle  will  also  range  with  it,  if  there  is  no 
variation,  but  if  the  north  point  of  the  needle  is  to  the  west  of  the 
sight-vane,  the  variation  is  westerly,  if  to  the  east,  easterly,  and  the 
number  of  degrees  and  parts  of  a  degree  that  the  needle  deviates 
from  the  direction  of  the  sight-vanes  shows  the  amount  of  the 
variation. 

But  it  is  not  to  be  supposed  that  any  particular  observer  can  be 
at  the  points  and  places,  where  the  sun  is  either  rising  or  setting 
just  at  the  time  the  sun  is  on  the  equator.  We  must  have  a  broader 
basis,  and  in  fact  by  means  of  the  latitude  of  the  observer  and  the 
decimation  of  the  sun,  any  observer  has  the  means  of  knowing  the 
precise  direction  in  which  the  sun  will  rise  or  set,  any  day  in  any 
year. 

Let  us  suppose  that  the  sun  on  a  certain  day,  observed  from  a 
certain  place,  must  have  arisen  S.  81°  E.,  but  by  the  compass  it  was 
observed  to  rise  S.  79°  JS.,  the  variation  of  the  compass  was  there- 
fore 2°  west. 

These  observations  are  called  taking  an  azimuth.  Azimuths  are 
often  taken  at  sea  to  determine  the  variation  of  the  compass. 

On  land,  however,  the  horizon  is  rarely  visible,  and  very  few  obser- 
vations on  sun  rise  or  sun  set  can  be  made,  besides  there  are  other 
objections  arising  from  atmospherical  refraction ;  it  is  therefore  best, 
most  convenient,  and  more  conducive  to  accuracy,  to  take  the  sun 
when  up  10,  15  or  25°  above  the  horizon,  and  observe  its  direction 
per  compass,  and  compare  the  result  to  the  computed  bearing  for 
the  same  moment,  and  if  the  two  results  agree  the  compass  has  no 
variation  ;  if  they  disagree  the  amount  of  such  disagreement  is  the 
amount  of  the  variation  of  the  compass. 

.,'  "\_    -        M  j~'  Y 

£    UNIVERSITY 

V  OF 


106  SURVEYING. 

By  means  of  spherical  trigonometry  the  true  bearing  of  the  sun 
can  be  determined  at  any  time,  on  the  supposition  that  the  observer 
knows  his  latitude,  the  declination  of  the  sun,  and  its  altitude  ;  these 
three  conditions  furnish  a  triangle  like  PZS. 

The  altitude  subtracted 
from  90°,  gives  ZS,  the  lati- 
tude from  90°,  gives  ZP,  and 
PS  is  found  by  adding  or 
subtracting  the  «eun's  declina- 
tion to  90°,  according  as  it  is 
north  or  south. 

ZS  is  co-altitude,  ZP  is 
co-latitude,  and  PS'is  the  sun's 
polar  distance  ;  the  angle  PZS 
is  required,  and  it  can  be 
found  by  the  following  rule. 

1.  Add  the  three  sides  of  the  triangle  together  and  take  the  half  sum. 
From  the  half  sum,  subtract  the  sun's  polar  distance,  thus  finding  the 
remainder. 

2.  Add  the  sin-complement  of  the  co-altitude,  the  sin-complement  of 
the  co-latitude,  the  sine  of  the  half  sum,  and  the  sine  of  the  remainder. 
The  sum  of  these  four  logarithms  divided  by  2,  will  be  the  cosine  of 
half  the  azimuth  angle. 

N.  B.  This  rule  is  the  application  of  equations  on  page  204 
Robinson's  Geometry.  The  sin-complement  is  the  logarithmic  sine 
of  an  arc,  subtracted  from  10. 

EXAMPLES. 

In  latitude  39°  6' 20"  north,  when  the  sun's  decimation  was  12° 
3'  10"  north,  the  true  altitude  of  the  sun's  center  was  observed  to 
be  30°  10'  40",  rising.  What  was  the  true  bearing  of  the  sun,  or  its 
azimuth  ? 

90°  90  90 

Lat.    39     6  20  Alt.    30.  10.  40        Dec.  12°  3'  10 

co-Lat.    60  5340        co-Alt.    59.  49.  20        PD    77.  56  60 


VARIATION    OF   THE    COMPASS.  107 

P.  D.     77°  56.  50 

co-Lat.  50.  53.  40  sin.  com.     0.110146 
co- Alt.  59.  49.  20  sin.  com.     0.063295 


2)188  39  50 

94  19  55  sin.         9.997758 
77  56  50 


Rem.    16  23  5  sin.         9.450376 


2)19.622575 
49°  38' 30"  cosine  9.811287 


Bearing,       99°  17'  0    from  the  north,  or  80°  43'  from  the 
south. 

If  at  the  time  of  taking  the  altitude  of  the  sun,  another  observer 
had  taken  its  bearing  by  the  compass,  and  found  it  to  be  S.  80°  43' 
E.,  then  the  compass  would  have  no  variation,  and  whatever  it  dif- 
fered from  that  would  be  the  amount  of  variation. 

If  a  line  were  run  along  the  ground,  direct  toward  the  center  of 
the  sun,  at  the  time  the  altitude  was  taken,  and  sufficiently  marked, 
that  would  be  a  standing  line  of  known  direction  ;  and  if  from  any 
point  in  that  line,  we  could  draw  another  line,  making  an  angle  with 
it  of  99°  17'  on  the  north,  or  80°  43'  on  the  south,  such  a  line 
definitely  marked,  would  be  a  permanent  meridian  line,  for  all  time 
to  come ;  on  which  we  could  at  any  tune  place  a  compass,  and 
observe  its  variation. 

Let  AS  be  the  line  toward  the  sun, 
along  the  ground,  AE  a  line  due  east,  and 
Mm  a  true  meridian  line.  The  angle  SAE 
must  equal  9°  17'. 

To  make  that  angle,  take  AS,  one  chain 
or  100  links  ;  from  S,  draw  the  line  SE 
at  right  angles  to  AS,  by  means  of  a 
surveyor's  cross.* 

From  S  take  SE,  of  such  a  value  as  will  make  SAE  9°  17', 
which  is  determined  by  trigonometry ;  as  follows, 

*  Surveyor's  cross  is  nothing  more  than  a  pair  of  sight  vanes,  set  at  right 
angles  with  each  other,  for  the  purpose  of  making  right  angles. 


108  SURVEYING. 

As  100.  :  £#=Rad.  :  tan.  9°  17' 
Whence        £g=100-     tan-  9°  ir 


JR. 

£#=16.347  links. 

That  is,  from  *S  measure  off  16  and  a  little  more  than  ^  of  a  link, 
and  there  is  the  point  E.  A  line  drawn  from  A  to  JK,  is  a  due  east 
and  west  line. 

If  we  put  the  surveyor's  cross  on  this  line,  at  any  point  as  A,  and 
range  one  branch  of  it  along  the  line  AE,  the  other  branch  will 
mark  out  the  line  Mm,  a  true  meridian,  if  everything  has  been  done 
to  accuracy. 

In  the  afternoon,  or  some  other  day,  another  meridian  may  in 
like  manner  be  drawn  near  this  one,  and  if  they  are  both  true  meri- 
dians, they  will  be  parallel.  If  not  parallel,  other  observations 
should  be  made  until  some  two  or  three  are  obtained,  that  are 
parallel  or  very  nearly  so  ;  and  the  mean  direction  then,  may  be 
regarded  as  the  true  meridian. 

A  true  meridian  will  always  be  a  test  line  for  a  compass  ;  and  by 
placing  any  compass  upon  it,  the  declination*  of  the  needle  can  be 
determined. 

Again.  In  the  triangle  PZS,  if  we  compute  the  angle  ZPS  (as 
is  done  on  page  211  Robinson's  Geometry),  we  shall  have  the  sun's 
distance  from  the  meridian  or  the  apparent  time  ;  then  if  we  have  a 
time  piece  that  can  be  relied  upon,  for  three  or  four  hours,  we  can 
determine  the  time  within  a  few  seconds,  when  the  sun  will  be  on 
the  meridian.  A  line  at  that  time,  run  direct  toward  the  center  of 
the  sun,  will  define  the  meridian. 

The  objections  to  these  methods  are, 

1.  The  sun  is  a  large  body,  and  its  center  cannot  be  exactly 
defined. 

2.  The  sun  changes  position  so  rapidly  that,  unless  we  are  in  an 
observatory,  where  every  thing  is  prepared  and  in  order,  it  is  diffi- 
cult to  get  observation  upon  it. 


*  Declination  of  the  needle,  in  common  language,  is  called  the  variation  of  the 
compass  ;  and,  as  a  general  thing,  we  adhere  to  common  language. 


VARIATION    OF   THE   COMPASS.  109 

3.  The  sun  is  so  bright  an  object  that  it  cannot  be  viewed  with- 
out prepared  glasses. 

4.  The  majority  of  persons  that  have  been,  and  probably  will  be 
practical  surveyors,  have  not  the  instruments  to  take  altitudes  of  the 
sun,  and  they  are  not  and  cannot  be  at  home  in  astronomical  obser- 
vations and  computations. 

Some  of  these  objections  are  deserving  of  little  respect,  and  others 
can  be  partially  removed. 

For  instance,  if  the  sun  is  too  large,  and  too  brilliant  to  be  accur- 
ately and  deliberately  observed,  we  can  take  the  planet  Venus,  Jupiter, 
or  Saturn,  and,  by  proper  observations,  determine  their  directions, 
during  the  twilight  of  evening,  when  we  can  see  the  planet  distinctly, 
and  at  the  same  time  that  other  objects  are  sufficiently  distinct  to  run 
lines.* 

But  the  method  most  known  and  most  in  favor  among  practical 
men,  is  that  of  taking  the  direction  of  the  north  star. 

The  north  star  is  a  star  of  the  second  magnitude  (Polaris),  whose 
right  ascension,  Jan.  1,  1851,  was  Ih  5m  18s  (at  present  increas- 
ing at  the  rate  of  17s  71  per  annum  ),  and  declination  was  then  88° 
30'  55",  with  an  annual  increase  of  19"8,  it  is  therefore,  but  1°  29' 
5"  from  the  pole,  and  it  is  called  the  pole  star  or  north  star  because 
it  is  so  near  the  pole. 

If  the  star  were  situated  directly  at  the  polar  point,  a  line  toward 
it  would  be  the  true  meridian  line,  but  being  1°  29'  5"  distant,  the 
star  apparently  makes  a  circle  round  the  pole  hi  a  siderial  day, 
making  two  transits  across  the  meridian,  one  above  and  the  other 
below  the  pole,  —  a  direction  to  it,  at  these  times,  would  be  a  true 
meridian  line. 

To  find  these  times,  subtract  the  right  ascension  of  the  sun  from  the 
right  ascension  of  the  star ;  increasing  the  latter  by  24h,  to  render 
the  subtraction  possible,  when  necessary. 

*  For  example,  in  the  year  1853,  from  the  25th  of  July  to  the  5th  of  August, 
the  planet  Jupiter  will  pass  the  meridian  in  the  evening  twilight.  On  the  first 
of  August,  Jupiter  will  pass  the  meridian  of  New  York,  at  8h  llm  53s,  and  it 
will  pass  the  meridian  of  Cincinnati,  at  8h  11  39s,  mean  local  time  ;  and,  of 
course,  whoever  is  able  to  designate  that  time  within  a  few  seconds,  and  is  also 
prepared  to  mark  the  direction  of  the  planet,  will  have  a  true  meridian  line. 

The  moon  is  not  a  good  object  for  this  purpose;  it  changes  its  place  too  rapidly. 


110  SURVEYING. 

The  difference  will  be  the  time  of  the  upper  transit,  and  llh  and 
59  minutes  from  that  time  will  be  the  time  of  the  lower  transit. 
The  right  ascension  of  the  sun  is  to  be  found  in  the  Nautical  Alma- 
nacs, for  every  day  in  the  year ;  and  it  is  nearly  the  same,  for  the 
same  day,  in  every  year. 

For  example.  At  what  times  will  the  north  star  make  its  transits 
over  the  meridian  on  the  first  day  of  July,  1863. 

H.  M.    s. 

*  &  A+24h        -         -         -        -         25     6     0 
0  R.  A. 6  41   16 


18  24  44 

This  result  shows  that  the  upper  transit  will  occur  about  6h  24m, 
hi  the  morning  of  the  2d  of  July.  I  say  about,  because  I  took  the 
sun's  right  ascension  for  the  morning  of  July  1,  and  from  that  time 
to  6,  next  morning,  is  1 8  hours  :  and  during  this  time  the  right  as- 
cension of  the  sun  will  increase  full  3  minutes, —  therefore  the 
upper  transit  will  take  place  6h  21m  in  the  morning,  and  the  pre- 
vious lower  transit  llh  59m  previous,  or  at  6h  22m,  evening. 

But  neither  of  these  transits  will  be  visible,  as  they  both  occur  in 
broad  day  light,  from  any  place  where  the  north  star  is  ever  dis- 
tinctly visible. 

In  summer,  then,  when  most  surveying  is  done,  the  meridian 
transits  of  the  north  star  are  not  visible,  nor  is  this  important :  for 
the  transits  are  seldom  used,  by  reason  of  two  objections  : 

1.  The  star  changes  its  direction  most  rapidly  while  passing  the 
meridian. 

2.  Observers,  generally,  have  not  the  means  of  knowing  the  time 
to  sufficient  accuracy.* 

To  obviate  these  objections,  observations  may  be  taken  on  the 
star  at  its  greatest  elongations  ;  for,  about  those  points  and  for  full 

*  NOTE. —  Very  few  persons  consider  that  their  clocks  and  watches,  however 
good  and  valuable,  do  not  give  the  exact  time,  but  only  approximations  to  the 
time. 

For  any  astronomical  purpose,  like  the  one  under  investigation,  the  charac- 
ter of  the  time  piece  should  be  well  tested —  its  rate  of  motion  known  — and 
its  errors  established  by  astronomical  observations. 


VARIATION    OF    THE    COMPASS. 


Ill 


15  minutes  before  and  after,  the  star  does  not  visibly  change  its 
direction  ;  hence  the  observer  has  a  sufficient  interval  to  be  deliber- 
ate, and  he  can  be  sufficiently  exact  as  to  time  without  any  extra 
trouble. 

The  following  tables  show  the  times  of  the  greatest  eastern  and 
western  elongations,  which  occur  in  the  night  season.  These  tables 
are  not  perpetual,  but  they  will  serve  without  correction  for  20  years 
or  more  to  come. 

EASTERN     ELONGATIONS. 


Days. 

April. 

May.      1     June. 

July. 

August. 

Sept 

H.   M. 

H.  M.    1     H.  M. 

H.  M. 

H.    MT 

H.  M. 

1 

18  18 

16  26 

14  24 

12  20 

10  16 

8  20 

7 

17  56 

1603 

14  00 

11  55 

9  53 

7  58 

13 

1734 

15  40 

1335 

11  31 

930 

7  36 

19 

17  12 

15  17 

13  10 

11  07 

9  08 

7  15 

25 

16  49 

1453 

1245 

1043 

8  45 

663 

WESTERN     ELONGATIONS. 


Days. 
1 

7 
13 
19 
25 

Oct. 

NOT. 

Dec. 

Jan.   :  Feb.  j  March. 

H.  M. 

18  18 
17  56 
17  34 
17  12 
16  49 

H.  M. 

16  22 
15  59 
1535 
15  10 
1445 

H.  M. 
14  19 

13  53 
1327 
1300 
12  34 

H.  M. 
1202 

11  36 
11  10 
10  44 
10  18 

H.  M. 

9  50 
926 
902 
8  39 
8  16 

H.  M. 

801 
738 
716 
6  54 
6  33 

It  will  be  observed  that  these  times  are  astronomical ;  the  day 
commencing  at  noon,  and  12h  40  means  40m  after  midnight,  etc. 

Now,  admitting  that  the  direction  of  the  star  can  be  observed,  the 
next  step  is  to  find  how  much  that  direction  deviates  from  the  meri- 
dian —  and  this  is  a  problem  in  spherical  trigonometry. 

A  great  circle  passing  through  the  zenith  of  the  observer  to  the 
star,  when  the  star  is  at  one  of  its  greatest  elongations  will  touch  the 
apparent  small  circle  made  by  the  apparent  revolution  of  the  star 
about  the  pole,  and  will  therefore,  with  the  star's  polar  distance,  form 
a  right  angle —  and  we  shall  have  a  right  angled  spherical  triangle, 
of  which  the  observer's  co-latitude  is  the  hypotenuse,  the  star's  polar 
distance  one  side,  and  the  angle  opposite  to  this  side  is  the  angle 
required. 


112 


SURVEYING. 


EXAMPLE. 


What  will  be  the  bearing  of  the  north  star  observed  from  latitude 
42°  N.  in  the  year  1860,  when  the  star's  polar  distance  will  be 
1°26'12"?  Ans.  1°  56'. 


As  cos.  Lat.  42°        - 

is  to  radius  - 

So  is  sin.  1°  26'  12" 


9.871073 

-  10.000000 

8.399183 


To  sin.      1°56'    -  -     8.628110 

In  this  manner  the  following  table  was  computed.  The  mean 
angle  only  is  put  down,  being  computed  for  the  first  of  July  in  each 
year. 

AZIMUTH    TABLE. 


Years. 

1852 

Lat.  30° 
Azimuth. 

Lat.  35° 
Azimuth. 

Lat.  40° 
Azimuth. 

Lat.  45° 
Azimuth. 

Lat.  50° 
Azimuth. 

1°  42'  30" 

1°48'21" 

1°  55'  52" 

2°  5'  32" 

2°  18'    5' 

1854 

1°41'45" 

1°47'39" 

1°55'    2" 

2°  4'  30" 

2°  17'    6' 

1856 

1°41'    2" 

1°  46'  49" 

1°54'12'/ 

2°  3'  44" 

2°  16'    9' 

1858 

1°  40'  27" 

1°46'11" 

1°  53'  30" 

2°  3'    2" 

2°  15'  12' 

1860 

1°  39'  43" 

1°45'24" 

1°52'32" 

2°  2'    4' 

2°  14'  16' 

1862 

1°  38'  50" 

1°44'29" 

1°51'44" 

1°!'    2" 

2°  13'  18' 

This  table  is  given  for  those  who  may  wish  to  use  it,  but  we 
would  recommend  each  observer  to  follow  the  example  which  pre- 
cedes the  table,  and  compute  the  azimuth  corresponding  to  his 
latitude  and  time. 

THE    PRACTICAL    DIFFICULTY. 

The  north  star  is  not  brilliant,  it  cannot  be  seen  until  it  is  so  dark 
that  all  minute  terrestrial  objects  are  totally  invisible,  it  is  therefore 
difficult  to  draw  a  line  and  accurately  mark  it.  All  these  night 
operations  are,  at  best,  perplexing  and  inaccurate,  yet,  by  the  means 
of  lights  and  artificers,  lines  can  be  drawn. 

If  the  observer  have  a  theodolite  and  an  assistant,  there  will  be  no 
difficulty.  Let  them  be  at  the  place  from  which  they  wish  to  take 
the  observation  in  time,  adjust  the  instrument  and  direct  the  telescope 
to  the  north  star.  Now  sufficient  light  must  be  reflected  into  the 
telescope  to  enable  the  observer  to  see  the  cross  hairs,  and  this  may 
be  done  by  the  assistant  holding  a  light  before  a  stiff  sheet  of  white 


TO  SURVEY  WITHOUT  A  COMPASS.  113 

paper,  so  as  to  throw  the  reflected  light  from  the  paper  into  the 
telescope,  or  this  may  be  done  by  means  of  a  stand  to  hold  both  the 
light  and  the  paper. 

When  the  vertical  spider's  line  becomes  visible,  let  the  star  be 
brought  directly  upon  it,  and  if  it  is  near  the  time  of  greatest  elon- 
gation it  will  appear  to  remain  so,  for  some  time.  But  if  the  star 
has  not  reached  it  greatest  elongation,  it  will  move  from  the  line 
more  to  the  east,  if  the  elongation  is  easterly,  and  more  to  the  west, 
if  westerly. 

The  telescope  must  be  continually  directed  to  the  star,  by  means 
of  the  tangent  screw  of  the  horizontal  plate,  but  for  some  tune  the 
spider  line  and  star  will  coincide  without  moving  the  screw,  and 
then  the  star  will  depart  from  the  line  in  the  contrary  direction  to  its 
former  motion,  but  the  telescope  must  no  longer  follow  the  star,  its 
position  will  now  show  the  direction  to  the  star,  when  the  star  had 
its  greatest  elongation,  and  thus  it  should  be  left  until  morning. 

In  the  morning,  carefully  range  and  mark  a  line  through  the 
telescope. 

If  we  now  make  an  angle  with  this  line  equal  to  the  azimuth,  by 
means  of  the  theodolite,  or  by  means  of  measuring  a  triangle  as 
explained  hi  the  former  part  of  this  chapter,  and  mark  this  new  line 
either  to  the  right  or  left,  as  the  case  may  require,  we  shall  then 
have  a  permanent  meridian  Kne  for  all  future  use. 

By  placing  a  compass  on  any  well  defined  and  true  meridian  we 
can  determine  its  variation  by  simple  observation. 

If  we  have  not  a  theodolite,  we  can  obtain  a  tolerably  accurate 
direction  to  the  north  star  by  means  of  illuminated  plumb  lines  sus- 
pended in  vessels  of  water,  so  placed  as  to  range  to  it. 


CHAPTER    IV. 

TO  SURVEY  WITHOUT  A  COMPASS. 

THE  inquiry  is  sometimes  made,  whether  lands  could  be  surveyed 
without  a  -compass  ;  we  reply  in  the  affirmative.  The  compass  is 
only  a  convenience,  and  if  it  had  never  been  discovered,  it  is  probable 


114  SURVEYING. 

that  surveys  would  have  been  more  accurately  made.  Too  much 
reliance  has  been  placed  on  the  accuracy  of  the  compass,  and  in 
consequence  little  attention  has  been  paid  to  defining  any  astronomi- 
cal lines. 

Were  it  not  for  the  compass,  it  is  probable,  that  every  country- 
town,  and  even  every  large  land  holder,  would  have  meridian  lines 
well  defined  about  his  premises. 

Having  a  meridian  line  to  start  upon,  we  can  find  angles  and 
define  the  position  of  lines  very  accurately  by  means  of  a 

CIBCUMPERENTOR. 

The  circumferentor  consists  of  a  horizontal  circular  plate  divided 
into  360  degrees,  over  which  an  index  bar,  or  another  circular  plate, 
is  made  to  revolve.  This  index  bar  carries  sight-vanes  or  a  telescope. 
The  index  bar  or  the  revolving  circular  plate  also  carries  a  vernier 
scale,  which  will  enable  the  operator  to  make  an  angle  to  one  minute 
of  a  degree.  The  whole  instrument  is  placed  on  a  tripod,  and  by 
the  aid  of  spirit  levels  attached  to  the  lower  plate,  the  horizontal 
position  is  attained  with  a  sufficient  degree  of  accuracy. 

The  figure  before 
us,  represents  the 
essential  parts  of  a 
circumferentor.  JVS 
is  considered  as  the 
primative  or  meri- 
dian line,  and  AE  is 
the  index  bar,  which 
turns  horizontally  on 
the  common  center. 

Vernier  scales  are 
fitted  into  the  index 
bar,  and  revolve  over 
the  graduated  arc. 
At  A  and  B  are 
openings,  to  receive 
cross  hairs  or  a  telescope. 

When  a  vertical  semicircle  is  made  to  revolve  vertically  through 


TO   SURVEY    WITHOUT   A   COMPASS.          115 

the  plane  AB,  and  the  diameter  of  that  circle  a  telescope,  then  we 
have  all  the  essentials  of  a  theodolite. 

To  most  theodolites,  a  magnetic  needle  is  attached,  but  the  mag- 
netic needle  is,  properly  speaking,  no  part  of  the  instrument. 

To  show  the  manner  of  finding  the  direction  between  two  given 
points,  by  means  of  the  circumferentor,  we  propose  the  following 
problem. 

Mr.  T.  H.  Jones  wishes  me  to  run  the  east  line  of  his  lot,  in  the 
town  of  A,  and  give  the  true  bearing,  the  corners  being  known.  In  the 
public  square  of  the  town,  about  one  and  a  quarter  miles  distantt  a 
meridian  line  has  been  established. 

Let  Mm  be  the  established  meridian  in 
the  public  square,  and  GHtiiQ  direction  of 
the  line  required. 

Place  the  circumferentor  on  the  meri- 
dian line  Mm,  so  that  NS  of  the  instru- 
ment will  coincide  with  it,  the  center  of 
the  instrument  being  at  a  in  a  road. 

The  general  direction  of  the  road  is  a  b, 
and  the  index  bar  AB  is  made  to  revolve  over  the  plate,  which  is 
firmly  fixed,  until  the  index  bar  or  sight  vanes  point  out  the  line 
ab.  The  line  is  run  by  means  of  ranging  objects  ;  such  as  flag  staffs, 
if  the  line  is  long ;  or  if  short,  by  sending  on  a  flag,  and  stationing 
it  at  b. 

Now  clasp  the  index  bar  on  the  plate,  by  the  clamp  screw  under 
it  (made  for  the  purpose).  Leave  a  flag  at  a  ;  take  the  instrument 
to  b,  and  there  place  it,  so  that  the  sight  vanes  will  range  back  to  a; 
then  the  position  of  NS  on  the  instrument  will  show  a  meridian  line 
through  that  point. 

Here  the  road  bends  a  little,  unclamp  the  index  (being  careful 
that  NS  rigidly  retains  its  position),  and  direct  it  to  the  general 
direction  of  the  road  be.  Mark  the  point  c,  by  an  object  as  before, 
and  mark  some  other  point,  so  as  to  secure  the  line  (the  other  point 
may  or  may  not  be  b).  Now  clamp  the  index  again,  and  remove  the 
instrument  to  c.  Place  the  instrument  firmly  as  before,  and  make 
the  index  range  along  the  line  be  ;  the  line  NS  of  the  instrument, 


116  SURVEYING. 

will  mark  out  a  meridian  line  at  the  point  c,  and  tlus  we  can  transfer 
the  meridian  line  Mm  to  any  other  point  whatever. 

Thus  we  may  go  to  any  point  d,  in  the  given  line  ;  no  matter, 
theoretically  speaking,  how  many  angles  we  have  made  during  the 
traverse.  Placing  the  instrument  at  d,  with  its  index  to  range  along 
the  last  line,  the  line  NS  of  the  instrument  gives  the  meridian^ 'mi 

Now  unclamp  the  instrument,  and  direct  Its  index  along  the 
required  line  OH ;  the  position  of  the  index,  on  the  graduated  plate, 
will  give  the  angle  from  the  north,  which,  by  means  of  the  vernier, 
can  be  determined  with  great  exactness. 

In  this  manner  we  may  go  to  any  point,  and  place  a  meridian 
there,  and  then  run  any  required  line  whatever ;  therefore,  we  can 
survey  any  field,  farm,  or  tract  of  land,  without  a  compass,  if  we 
have  a  circumferentor,  and  a  meridian  line. 

It  would  not  be  safe  to  transfer  meridians,  as  we  have  just  done, 
over  any  very  great  extent  of  country,  for  at  every  angle,  small 
errors  might  be  made,  and  the  accumulation  of  many  small  errors 
may  produce  too  great  inaccuracies  to  be  tolerated  or  overlooked. 
When  using  the  magnetic  needle,  no  errors  accumulate,  for  every 
setting  of  the  compass  is  primary,  and  independent  of  every  other. 

Therefore,  in  case  no  compasses  were  in  existence,  primary  meridi- 
ans, astronomically  established,  would  be  necessary  in  every  town  ; 
and  it  would  be  better  to  have  several  of  them  hi  the  same  town. 

From  the  foregoing  illustrations,  we  perceive  that  surveying  can 
be  done,  and  well  done,  without  a  compass,  yet  the  compass  is  an 
inestimable  blessing  to  mankind  ;  for  it  is  the  only  index  to  direction 
over  the  wild  waste  of  waters,  when  the  heavens  are  obscured,  and 
no  mariner  would  dare  brave  the  ocean  without  it. 


CHAPTER    V. 

ORIGINAL  AND   SUBSEQUENT  SURVEYS.— 

DIFFICULTIES  AND  DUTIES  OF 

A  SURVEYOR. 

IN  this  country,  lands  were  ceded  to  States,  or  sold  to  companies 
in  large  tracts,  without   any  definite  surveys;    the  boundaries 


SURVEYS    AND   SURVEYORS.  117 

described,  were  mountains,  rivers,  or  a  certain  number  of  miles 
along  the  shores  of  a  lake,  and  then  a  certain  number  of  miles  back. 
The  land  companies  hired  surveyors  from  time  to  time,  to  survey 
off  their  lands,  into  lots  of  100,  200,  and  600  acres ;  and  wherever 
these  surveyors  left  monuments  for  the  corner  of  lots,  established  the 
corners  for  all  time  to  come,  whether  correctly  placed  or  not. 

These  surveys  were  very  loose  and  inaccurate ;  it  could  not  be 
otherwise,  for  a  company  of  surveyors  would  frequently  run  15 
miles  hi  a  day ;  when  to  run  a  line  accurately,  and  measure  it,  four 
miles  is  a  good  day's  work. 

But,  notwithstanding  inaccuracies,  these  surveys  are  legal  and 
cannot  be  changed ;  "  thou  shalt  not  move  thy  neighbor's  ancient 
land  mark,"  and  it  is  right  it  should  be  so  ;  for  any  attempt  at  cor- 
rection, would  create  more  trouble,  confusion,  and  injustice,  than 
it  could  remedy. 

Lots  originally  sold  for  100  acres  in  the  state  of  New  York, 
generally  contain  from  101  to  106  acres,  in  consequence  of  the  orig- 
inal surveyors  having  directions  to  have  their  lots  hold  out.  Where 
the  lots  thus  overrun  in  one  portion  of  the  tract,  they  fall  short  on 
another,  for  the  surveyors  were  probably  desirous  to  show  to  the 
company,  that  their  grant  actually  contained  as  much  land  as  was 
anticipated. 

The  author  surveyed  one  of  these  lots,  that  originally  sold  for  100 
acres,  and  found  that  it  contained  but  a  little  over  76  acres. 

Some  of  the  companies  had  their  grants  laid  off  into  townships 
6  miles  square,  or  6  by  8  miles  ;  then  each  township  into  four  sec- 
tions, each  section  divided  off  into  lots,  and  the  lots  numbered, 
generally  beginning  at  the  south-west  corner. 

The  description  of  the  lots  hi  the  deeds  given,  were  very  loose 
and  indefinite,  stating  the  township,  section,  and  number  of  the  lot, 
containing  100  acres,  "  be  the  same  more  or  less,"  and  in  some  lots 
it  was  more,  and  hi  other  lots  it  was  less. 

As  we  before  remarked,  any  land  mark  to  the  corner  of  a  lot  laid 
down  by  these  original  surveyors,  must  remain  ;  subsequent  sur- 
veyors can  straighten  lines  between  point  and  point,  and  decide  what 
the  true  courses  are,  and  how  many  acres  the  lot  contains. 

When  a  surveyor  is  called  to  survey  any  farm  or  estate  that  has 


118  SURVEYING. 

been  previously  surveyed,  lie  must  find  some  corner  as  a  place  of 
^commencing,  and  from  thence  run  a  random  line,  as  near  the  true 
line  as  his  judgment  permits  ;  and  if  he  strikes  another  corner  he  has 
run  the  true  course,  if  not,  he  corrects  his  course,  as  taught  in  chap- 
ter II.  Thus,  he  must  go  round  the  field  from  corner  to  corner. 
He  has  a  right  to  establish  corners  only  where  no  corners  are  to  be 
found,  and  no  evidence  can  be  obtained  as  to  the  existence  and 
locality  of  a  former  land  mark. 

It  may  be  the  case,  that  a  surveyor  is  called  to  survey  a  lot  where 
no  corners  are  to  be  found.  If  a  fence  or  line  exists,  which  has 
been  the  undisputed  boundary  for  a  long  time,  that  boundary  can- 
not be  changed,  and  the  surveyor  must  establish  a  corner  by  ranging 
some  other  line  to  meet  the  first.  Sometimes  corners  may  be  found 
to  some  neighboring  lot,  from  which  lines  can  be  run,  to  establish  a 
corner  to  the  lot  we  wish  to  survey. 

Lines  of  lots  hi  the  same  town,  are  generally  parallel,  and  a  sur- 
veyor who  offers  his  services  to  the  public,  must  make  himself 
acquainted  with  the  general  directions  of  the  lines  of  lots,  over  that 
section  of  country  where  his  services  are  required. 

When  a  surveyor  is  called  to  divide  a  piece  of  land,  he  is  then  an 
original  surveyor,  and  not  liable  to  be  embarrassed  by  old  lines  and 
old  traditions,  he  has  then  only  his  mathematical  problem  before 
him. 

Owing  to  the  inaccuracies  of  original  surveys,  and  the  impossi- 
bility of  leaving  proper  land  marks,  hi  consequence  of  the  great 
haste  in  which  lands  were  originally  surveyed ;  great  confusion  has 
followed,  in  some  sections  of  our  country,  hi  respect  to  lines,  and  it 
has  been  no  uncommon  thing  to  have  whole  neighborhoods  at  vari- 
ance, if  not  in  law,  in  reference  to  the  boundaries  of  their  lands. 

In  cases  of  this  kind,  one,  and  then  another  of  the  disaffected, 
have  successively  employed  surveyors,  and  surveyors  thus  employed, 
are  apt  to  act  the  part  of  advocates,  rather  than  arbitrators,  and 
survey  too  much  according  to  the  direction  of  their  employer  ;  but 
all  such  efforts  to  settle  difficulties,  but  aggravate  them  more  and 
more. 

On  the  contrary,  however,  if  the  surveyor  clearly  understands  his 
duties,  and  can  rise  above  being  a  special  advocate  for  any  one  of 
the  parties  concerned,  he  can  do  more  than  judges  or  juries  to  restore 


SURVEYS    AND    SURVEYOR'S.  H9 

harmony  and  peace.  To  illustrate  these  views,  and  possibly  to  give 
some  valuable  instruction  to  some  readers,  we  give  a  history  of  a  case 
of  this  kind,  which  occured  in  the  year  1837,  in  the  county  of 
Ontario,  hi  the  State  of  New  York.  A  tract  of  land  consisting  of 
about  670  acres>  of  an  irregular  shape,  was  divided  on  paper  into 
five  equal  parts,  and  sold  to  five  different  individuals. 

The  whole  670  acres  was  bounded  by  four  lines,  no  two  of  them 
were  equal,  and  neither  of  the  angles  was  a  right  angle.  The 
largest  boundary  line  could  not  be  directly  measured  on  account  of 
an  impassable  ravine  ;  and  the  banks  of  this  ravine  was  so  thickly  set 
with  hemlocks,  that  it  was  impossible  even  to  sight  across. 

In  consequence  of  the  irregular  shape  of  the  whole,  and  the 
impossibility  of  directly  measuring  the  principal  boundary,  they  had 
never  been  able  to  agree  on  their  division  lines. 

Each  one  imagined  that  his  neighbor  was  inclined  to  crowd  upon 
him,  and  although  permanent  fences  were  desirable,  none  could  be 
made  until  lines  were  agreed  upon.  They  had  employed  several 
surveyors,  but  they  had  not  been  able  to  agree  on  their  divisions. 
In  this  state  of  things  a  surveyor  was  called  upon  to  go  and  make 
a  division  of  this  land,  but  the  difficulties  of  so  doing  were  carefully 
concealed  from  him. 

When  he  arrived  on  the  ground,  ready  for  operations,  the  whole 
neighborhood  was  present,  and  by  unmistakable  signs  he  soon 
learned  that  an  unusual  degree  of  interest  was  taken  in  the  survey. 

He  also  found  that  the  chief  difficulty  arose  from  not  being  able 
to  measure  the  line  CD.  All  the  corners,  A,  B,  C,  and  D,  were 
established.  The  surveyor  commenced  at  C  to  run  a  random  line 
as  near  CD  as  possible.  After  going  a  few  chains,  he  came  to  the 
bank  of  the  ravine  at  F,  where  it  was  impossible  to  pass  or  sight 
across.  Driving  a  stake  at  F,  he  took  a  direction  FH  along  the 
bank  of  the  ravine,  carefully 
noting  the  angle,  and  meas- 
uring the  line  to  H,  a  point 
where  objects  were  clearly  to 
be  seen  on  the  other  side  of  the 
ravine.  The  surveyor  then 
sent  a  man  over  with  a  flag, 
stationing  his  staff,  first  at  Kt 


120  SURVEYING. 

then  at  Z,  carefully  noting  the  direction  of  each,  and  being  careful 
to  have  the  angle  KHL  greater  than  30°.  He  then  passed  over  and 
set  the  compass  at  3C,  took  the  direction  of  KL  and  measured  it. 
Having  now  KL  one  side,  and  all  the  angles  of  the  triangle  HKL, 
he  computed  HL.  He  now  set  the  compass  at  L  and  took  a  definite 
direction  Lm ;  this  definite  direction  gave  him  the  angle  ffLm,  and 
he  now  had  all  the  angles  of  the  quadrilateral  ImFH,  and  two  of 
its  sides.  Whence  he  computed  the  exact  distance  to  m,  to  strike 
the  line  CF  produced. 

He  measured  that  distance  and  drove  a  stake  at  m,  and  computed 
mF.  The  company  now  ran  through  the  random  line,  driving 
stakes  at  the  end  of  every  third  chain,  and  the  random  line  came 
out  within  a  few  feet  of  the  established  corner  at  D.  The  surveyor 
measured  the  perpendicular  distance  to  D,  and  corrected  the  course 
by  the  rule  in  chapter  II.  He  also  computed  how  far  each  stake 
that  had  been  placed  on  the  random  line  must  be  moved  to  transfer 
it  to  the  true  line  ;  this,  the  reader  will  perceive,  was  done  by  propor- 
tional triangles. 

At  D  he  set  the  compass,  and  carefully  noted  the  course  and  dis- 
tance to  A.  He  then  returned  to  C,  taking  care  not  to  pass  along  the 
line  AB. 

At  C  he  set  the  compass,  and  carefully  noted  the  course  and 
distance  to  B.  He  now  computed  the  course  and  distance  from 
B  to  A.  The  line  lay  in  the  open  fields,  over  tolerably  smooth  ground, 
and  it  could  be  directly  and  accurately  measured. 

The  surveyor  now  called  all  the  parties  interested,  including  the 
sour  and  the  belligerent,  and  told  them  that  the  distance  from  B  to  A 
was  a  certain  number  of  chains  and  links,  and  that  they  would  now 
measure  it,  and  if  they  found  it  to  correspond  without  any  material 
error,  they  must  then  be  convinced  that  he  had  obtained  the  true 
length  of  CD,  and  that  he  could  then  divide  the  land  into  five  equal 
parts,  as  required. 

To  this  test  they  all  cheerfully  assented ;  the  line  was  measured 
and  corresponded  to  the  computation  within  three  links  ;  all  parties 
were  satisfied,  and  thus  ended  a  neighborhood  quarrel  of  six  years' 
standing. 

Previous  surveyors  commenced  at  the  point  D  and  run  DA,  AB9 


SURVEYS    AND    SURVEYORS.  121 

and  BC,  and  then  computed  CD.  This  was  more  direct,  simple,  and 
proper,  than  the  method  just  described,  but  it  left  no  test  behind 
it,  and  it  is  vain  to  expect  that  the  mass  of  men  will  receive  theoreti- 
cal computation  as  actual  measurement. 

Here,  and  in  most  other  cases  that  involve  contention,  the  surveyor 
must  not  only  convince  himself  that  the  survey  is  correctly  made, 
but  he  must,  if  possible,  show  others  that  his  conclusions  are  not 
only  right,  but  cannot  be  wrong ;  hence  judicious  surveyors  must 
often  measure  lines,  where  there  is  no  mathematical  necessity  for  so 
doing. 

The  next  duty  of  this  surveyor  was  to  divide  the  land  into  five 
equal  parts.  Each  one  had  previously  purchased  his  part,  and  he 
knew  its  locality,  but  not  his  exact  boundary  line  on  the  division. 

As  CD  was  not  parallel  to  AB,  and  AD  not  exactly  parallel  to 
CB,  to  divide  this  mathematically  exact  was  a  problem  of  considerable 
difficulty,  and  this  will  be  explained  in  the  next  chapter ;  but  practi- 
cally we  need  not  apply  all  mathematical  rigor,  the  surveyor  can 
divide  this  more  strictly  conformable  to  justice  without,  than  with 
the  mathematical  rigor. 

The  persons  who  had  the  two  most  eastern  lots,  had  the  worthless 
part  of  the  land  in  the  ravine,  and  of  course  if  any  one  had  an  excess 
of  area  it  should  be  those. 

To  find  where  or  nearly  where  the  divisions  come,  divide  the  line 
AB  into  five  parts  and  suppose  Q-  one  of  those  parts.  Now  BG  is 
not  quite  long  enough,  because  the  field  is  a  little  narrower  at  this 
end  than  at  the  other ;  the  surveyor  took  a  distance  BG  a  few  links 
greater  than  one  fifth  of  AB,  and  from  that  point  run  a  line  in  a 
medium  direction  between  BC  and  AD,  and  then  computed  its  area, 
the  result  would  show  whether  the  area  was  too  great  or  too  small, 
and  if  it  were  within  a  very  small  fraction  of  the  area  required,  the 
line  is  left  as  the  true  one,  otherwise  it  is  moved  as  the  case  requires. 
In  the  same  manner  the  other  division  lines  were  run. 

UNITED    STATES'    LANDS. 

Soon  after  the  organization  of  the  present  government,  several  of 
the  States  ceded  to  the  United  States  large  tracts  of  unoccupied 
land,  and  these,  with  other  lands,  since  acquired  by  treaty  and  pur- 
chase, constitute  what  is  called  the  public  lands. 


122  SURVEYING. 

Previous  to  1802,  there  was  no  general  plan  for  surveying  the 
public  lands,  or  in  fact,  no  surveys  were  made,  and  when  grants 
were  made  the  titles  often  conflicted  with  each  other,  and  in  some 
cases  different  grants  covered  the  same  premises. 

In  the  year  1802,  Colonel  I.  Mansfield,  then  Surveyor  General 
of  the  north-western  territory,  adopted  the  following  method  : 

Through  the  middle,  or  about  the  middle  of  the  tract  to  be  sur- 
veyed, a  meridian  is  to  be  run,  called  the  principal  meridian.  At 
right  angles  to  this,  and  near  the  middle  of  it,  an  east  and  west  line 
is  to  be  run,  and  called  the  principal  parallel. 

Other  meridians  are  to  be  run,  six  miles  distant  from  the  prin- 
cipal meridian,  both  east  and  west. 

Also,  parallels  of  latitude  are  to  be  run,  six  miles  from  the  prin- 
cipal parallel,  both  north  and  south. 

When  this  was  done  (  and  it  has  been  on  all  the  public  lands 
east  of  the  Mississippi  river  ),  the  whole  country  is  divided  into 
squares,  six  miles  on  a  side,  called  townships. 

Each  township  contains  36  square  miles.  Each  square  mile  is 
called  a  section,  and  it  contains  640  acres.  Sections  are  divided  into 
half  sections,  quarter  sections,  and  eighths.  But  these  divisions  are 
only  made  on  paper. 

When  a  person  makes  a  purchase  of  a  half  or  quarter  section,  it 
is  supposed  that  he  will  find  it  himself,  or  employ  a  surveyor  to 
mark  it  out. 

Townships  which  lie  along  a  meridian,  are  called  a  range,  and 
numbered  to  distinguish  them  from  each  other. 

Sections  are  regularly  numbered  in  every  township,  and  to  desig- 
nate any  particular  one,  we  say,  section  13,  in  township  number  4 
north,  in  range  3  east. 

This  shows  that  the  third  range  of  townships  east  of  the  princi- 
pal meridian,  in  township  No.  4  north  of  the  principal  parallel, 
is  the  township,  and  the  thirteenth  section  of  this  township  is  the  one 
sought. 

Not  more  than  ten  townships  north  or  south  of  a  principal  par- 
allel should  be  drawn,  before  a  new  principal  parallel  should  be 
designated,  and  new  measures  made  between  meridians :  because 
meridians  tend  toward  the  pole,  and  the  north  lines  of  townships 


DIVISION  OF  LANDS.  123 

will  be  theoretically  shorter  than  south  lines,  if  the  meridians  are 
run  by  the  compass. 

Where  the  public  lands  extend  to  rivers  and  lakes,  there  will  be 
fractional  townships  along  the  shores. 

Where  the  locality  of  a  particular  number  is  found  to  be  occupied 
by  a  lake  or  pond,  the  sale  is  void. 


CHAPTER    VI. 

METHODS. OF  SURVEYING  IRREGULAR 
FIGURES  AND  OF  DIVIDING  LANDS. 

FARMS  and  tracts  of  lands,  wholly  or 
partially  bounded  by  water,  as  represented 
in  the  figure  before  us,  are  surveyed  and 
there  areas  determined  by  drawing  right 
lines  within  the  tract  as  near  the  real 
boundaries  as  possible,  and  from  these  right 
lines,  at  equal  intervals,  measuring  the  off- 
sets to  the  real  boundary.  These  off-sets 
form  the  parallel  sides  of  trapezoids,  and 
as  they  are  all  equally  distant  from  each 
other,  the  computation  of  the  areas  they 
occupy  will  be  very  easy.  A  summary 
rule  for  finding  the  united  area  of  all  these  trapezoids  that  are 
bounded  by  one  line,  is  to  be  found  in  Prob.  VIII,  Mensuration. 
The  area  of  the  right  lined  figure  ABCDEFG,  is  found  as  directed 
in  Chapter  IV,  to  which  add  the  area  of  all  the  trapezoids,  and  we 
shall  have  the  area  of  the  whole. 

We  have  now  investigated  every  possible  case  of  computing  areas, 
and  we  are  now  prepared  to  divide  them.  Commencing  with  the 
most  simple  case  of  the  most  simple  figure,  the  triangle,  or  rather 
the  figure  that  has  the  least  number  of  sides. 


124  SURVEYING. 

PROBLEM    I. 
To  divide  a  triangle  into  two  parts,  having  a  given  ratio  of  m  ton. 

CASE  1.  By  a  line  drawn  from  one  angle 
to  its  opposite  side. 

Let  ABC  represent  the  triangle;  divide 
its  base  into  two  parts,  corresponding  to  the 
given  ratio,  and  let  AD  be  one  of  the  parts; 
then  we  shall  have  the  following  proportion. 

AD  :  AB  :  :  m  :  m-j-w 
Whence,     AD=~(AB)    and  BD^ 

Now  the  two  parts  are  numerically  known,  and  are  to  each  other  as 
m  to  n.  Triangles,  having  the  the  same  altitudes,  are  to  one 
another  as  their  bases.  Therefore,  ADO  :  CDB  :  :  m  :  n  as  re- 
quired. 

CASE  2.  By  a  line  parallel  to  one  of  its 
sides. 

Let  DE  divide  the  triangle  as  required, 
and  as  similar  triangles  are  to  one  another 
as  the  squares  of  their  homologous  sides, 
therefore  : 

(AB)*  :  (AD)2  :  :  m+n  :  m 


Whence,     An=AE^        m 

m-\-n 

Which  shows  that  if  we  have  the  numerical  value  of  AB,  and  of 
n  and  m,  we  can  find  that  of  AD,  and  from  D  draw  DE  parallel 
to  BO,  and  the  triangle  is  divided  as  required. 

CASE  3.  By  a  line  parallel  to  a'  given  line,  or  by  a  line  running  in 
a  given  direction. 

To  make  this  case  clear,  we  commence  by  giving  a  definite  ex- 
ample : 

There  is  a  triangular  piece  of  land,  from  one  of  the  angular  points, 
Ay  one  line  runs  N.  25°  W.,  distance  12  chains  ;  another  from  the  same 
point  runs  N.  42°  E.,  distance  15  chains.  It  is  required  to  divide  this 


DIVISION    OF    LANDS. 


125 


triangle  into  two  parts  in  the  ratio  2  to  3,  by  a  line  running  due  east 
and  west. 

Let  ABC  be  the  given  triangle,  and  B' C' 
the  required  division  line.  It  is  required  to 
find  the  numerical  value  of  A  C'  or  AB',  to 
make  the  area  AB'C'  f  of  the  area  ABC. 

Let  b  represent  the  side  of  the  triangle 
opposite  B,  and  c  the  side  opposite  C. 

Let  A  C'=x.  As  A  C'  and,  C'B'  have  defi- 
nite directions,  the  angle  AC'B'  is  given, 
also  AB'C'  is  given.  AC'B'=48°,  AB'C' 
=65°,  BAC=Q7°. 

In  the  triangle  AB'C'  we  have 

sin.  65°  :  x  :  :  sin.  48°  :  AB' 

sin.  48° 
Whence  AB  = 


Now,  by  Prob.  Ill,  Mens.,  area  ABC=\bc  sin.  A. 

sin.  48°  \ 


Also, 


area  AB'  C'=- 


sin.  65°  / 


0) 


sin.  A 


By  the  conditions  of  our  problem,  we  have  the  following   pro- 
portion. 

/sin.  48° 
|*MfcM:Hj 


sr" 


Or' 


sin.  48° 
*c:sin.65°*° 


5:2 


(2) 


We  may  here  stop,  and  make  the  problem  general. 

If  B'  C'  is  given  in  direction,  the  angles  B'  and  C'  will  be  given. 

We  now  require  the  division  of  the  triangle  ABC  into  two  parts, 
in  the  ratio  of  m  to  n  by  a  line  opposite  to  the  angle  A,  running  in 
a  given  direction. 

Represent  the  sides  of  the  given  triangle  adjacent  the  angle  A 
by  b  and  c,  b  extending  from  A  to  C,  and  c  extending  from  A  to  B. 

Put  #==  the  distance  from  A  to  the  division  line,  on  the  side  CA. 

Then,  by  the  preceding  proportion  we  have, 


126  SURVEYING. 

sin.  C" 


:  m 


Observe  that  a;  is  opposite  the  angle  J3',  the  sine  of  which  stands 
in  the  numerator  of  the  second  fraction.  Had  x  represented  AJB', 
sin  0'  would  have  been  the  numerator. 

Drawing  out  the  result  for  (2),  we  find  that 

5  sin  48° 
sin 


72  sin  65° 


CASE  4.  By  a  line  that  shall  pass  through  a  given  point  within 
the  triangle. 

A  point  in  a  triangle  cannot  be  given,  unless  the  perpendicular 
distances  from  that  point  to  the  sides  are  given,  and  if  these  per- 
pendicular distances  are  given,  then  we  can  readily  find  the  three 
distances  from  the  angular  points  of  the  triangle,  and  the  angles 
which  these  lines  make  with  the  sides 
of  the  triangle  are  known.  For  instance, 
if  the  point  P,  in  the  triangle  ABC,  is 
known,  PR  and  PT  are  known,  and 
all  the  angles  of  the  quadrilateral 
ATPR  are  known.  These  are  suffi- 
cient data  to  compute  the  line  AP,  and 
the  angles  RAP  and  TAP. 

REMARK.  —  We  may  now  require  a  triangle  to  be  cut  off,  by  a  line  running 
through  P,  which  shall  contain  any  definite  portion  of  the  triangle  ABC,  not 
involving  an  impossibility. 

For  instance,  if  the  point  P  is  near  the  center  of  the  triangle,  it  would  not 
do  to  require  us  to  cut  off  a  tenth  part  of  the  triangle,  or  any  smaller  portion, 
for  it  would  be  impossible  to  do  so.  When  it  is  required  to  cut  off  a  very  small 
portion  of  the  whole  triangle,  the  point  P  must  be  near  one  of  the  sides,  or 
near  one  of  the  angular  points.  Sometimes  the  required  quantity  can  be  cut 
off  from  one  angular  point,  sometimes  from  another,  and  sometimes  from  all 
three. 

Let  us  now  require  one-third  of  the  triangle  cut  off,  by  a  line  passing  through 


DIVISION   OF   LANDS.  127 

P,  taking  the  angular  point  At  and  let  EF  be  that  line.    We  are  to  determine  the 
value  of  AF. 

In  the  triangle  ABC,  the  angles  A,  B,  and  C,  are  known,  and  the 
sides  opposite  to  them,  a,  b  and  c,  are  also  known.  AP  is  known,  and 
call  it  k.  Put  the  angle  EAP=p,  PAF=q.  Then,  A=p-\-q. 
Put  AF=xt  AE=y. 

By  Prob.  III.  Mens.        area  ABC=$c  sin.  A 
Also  "  "  area  AJEF=%xy  sin.  A 

By  the  conditions  of  the  problem, 

\xy  sin.  A=^bc  sin.  A 
Whence,  ^xy—bc  (I) 

The  triangle  AFE  consists  of  two  parts.  AFP,  APE;  therefore, 
\hx  sin.  q-^-^hy  sin.  p=^xy  sin.  A 

Or  a?  sin.  q-\-y  sin.  jo^^-l!1?!  (2) 

If  we  had  required  the  nth  part  of  the  triangle  ABC,  in  place  of 
the  3rd  part,  equation  (1)  would  have  been  nxy=bc. 

Making  this  supposition,  to  make  the  problem  more  general,  we 

have#y=  -   and  y=  — .    By  the  aid  of  these  last  two  equations, 
n  nx 

(2)  becomes 


nx  nh 

Qr  , be  sin.  A^__ be  sin,  p 

nh  sin.  q  nsm.  q  (3) 

Whence,        XJ"*™'A±(  b*c*^A  J™±P\t 
Qnhsm.q     \4)i2h2s'm.2q     nsm.  q/ 

In  case  n  is  large,  that  is  the  part  to  be  cut  off  small,  the  value 
of  x  may  be  imaginary,  corresponding  to  the  preceding  remark. 

CASE  5.  When  the  given  point  is  on  one  side  of  the  triangle. 

The  two  parts  must  be  equal,  or  one 
of  them  will  be  less  than  half  of  the 
whole. 

We  always  compute  the  less  part.  Let 
P  be  the  point  in  the  line  AB,  PQ  and 
PR  perpendiculars  to  the  other  sides,  are 
known  ;  B  C  and  A  O  are  both  known. 
Now,  through  the  given  point  Pt  it  is 


128  SURVEYING. 

required  to  draw  PZ>,  so  that  the  triangle  BPD,  shall  be  the  nth 
part  of  ABC.    That  is 


Whence, 

CASE.  6.  When  the  given  point  is  without  the  triangle. 

Let  ABC  be  the  given  triangle,  and  P  any  given  point  without 
it.  It  is  required  to  run  a  line  from  Pt  to  cut  off  a  given  portion  of 
the  triangle  ABC,  or  (which  is  the  same  thing)  to  divide  the  tri- 
angle into  two  parts  having  the  ratio  of  m  to  n.  Let  PG  be  the 
line  required. 

As  P  is  a  given  point,  AP  is  a  line 
given  in  distance  and  position  ;  therefore, 
the  angle  PAH  is  known. 

Solution.—  Put  the  angle  PAff=u, 
CAB=v;  then  PAG=(u+v).  Also 
put  AG=x,  AH=y,  AP=a,  and  the 
area  of  the  triangle  AHQ-=mc,  me  being  a  known  quantity. 

Now,  (by  Prob.  III.  Mens.)  \xy  sin.  v=mc  (1) 

Also  "  "      \ax  sin.  (^4-v)=area  APG 

And  "  "       \ay  sin.  w=area  PAH 

Therefore,       %ax  sin.  (u+v)  —  \ay  sin.  u=mc  (2) 

Or,  x  sin.  (w-f  v)—y  sin.  v==?_^f  (3) 


a 

From  (1),  we  find  y—-^^L- which  value  substituted  in  (3)  gives 
asm.  v 

.     /         x     2  me  sin.  u     2  me 

x  sin.  (w+t>) — _ = 

x  sm.  v          a 

Whence,       *2sin.  (u+v)~  ^^=2mcsin^ 

a  sin.  v 

-  ...  2  me          2  me  sin.  w 


a  sn.  «--^          sn.  v  sn. 


m,       f  f         MS-  i,-          i       2  me  sin.  w 

Therefore,«^ 


in.  v  sin.  («-J- v) 


DIVISION    OF   LANDS  129 

EXAMPLES. 

1.  In  the  triangle  ABC,  the  side  AB=23.645  chains,   AC=\l.b\ 
chains,  and  J?  (7=  12. 575  chains. 

The  given  point  P  from  the  angle  A,  is  distant  10  chains,  at  an 
angle  of  40°  from  the  line  AC. 

It  is  required  to  draw  a  line  from  this  given  point  P,  through  the 
triangle,  so  as  to  divide  it  into  two  equal  parts. 

Whereabouts  on  AB  will  PGf  intersect  ? 

The  angle  £AC=31°  17'  19"=v.     PAH=40°=u.     Therefore 
PAG=7l°  17'  10"=(u+v.) 

The  area  of  the  triangle  AB  C  is   107.52  square  chains.     The 
part  to  be  cut  off  by  the  triangle  AEG  is  therefore  =  53.76 =m. 

We  must  use  the  natural  sines,  or  the  logarithmic  sines  if  we  omit 
them  in  the  index. 

me  log.-         -  1.730464 

asin.(w+v)log.     -         -         -       0.976406 

mc        .=5.676  log.     0.754058 


a  sin.  (u-^-v) 

2 

5!f! -=32.22          log.     1.508116 

a2  sm.2(u-\-v) 

2mc               log.  -         -         -        2.031494 

sin.«             log.  -         -        —1.808067 

27ttcsin.  u     -  ...        1.839561 

sin.  v  sm.(u-}-v)  -         -        — 1.691866 


14a6, 


sin.  v  sin.  (u-}-v) 

The  part  of  the  formula  under  the  radical  is  therefore  (32.22-f- 
110.51)  or  172.73. 

Whence  z=5.676db,/l  72.73=18.819,  or  —7.467. 

REMARK.  —  The  minus  sign  means  opposite  in  direction,  and  if  we  pro- 
duce AC  and  AG  (see  last  figure,)  to  the  left  of  A.  a  line  drawn  from  P 
through  a  point  which  is  (7.467)  to  the  left  of  A,  will  form  a  triangle  below 
the  line  AG,  which  will  be  equal  to  AHG. 
9 


130 


SURVEYING 


2.  We  have  a  right  angled  triangle  whose  base  is  47.87  chains,  and 
perpendicular  54.46  chains.  From  a  given  point  without  it,  we  are 
required  to  run  the  center  of  a  straight  road,  to  leave  one  sixth  of  the 
triangle  on  one  side  and  jive  sixths  on  the  other. 

From  the  acute  angle  at  the  base,  the  distance  to  the  given  point  is 
20  chains,  and  the  line  to  it  makes  an  angle  with  the  hypotenuse  of  30°. 

REMARK.  —  If  P  is  a  giveu  point,  its  distance 
and  direction  from  one  of  the  angular  points 
must  be  given  ;  and  if  the  distance  and  direction 
from  one  of  the  angular  points  is  given,  the  dis- 
tances and  directions  from  all  of  them  are  virtually 
given  ;  thus,  if  we  have  AP,  AC,  and  the  angle 
PAC,  we  have  CP,  and  the  angle  ACP,  and  we 
may  theorize  on  the  triangles  PCH,  CHG'  as  well 
as  on  APR  and  AHG. 


The  area  of  the  triangle  AB C— 13 04.94  square  chains. 
One  third  of  this  is  me—   217.485. 

—.^-—^go  J2'  20"      PA O^~ u " ' 

20.     AH=y.    AG=x. 

me  log.  - 

a=20          log.         1.301030  ) 
sin.  (u  +  v)  log.     —1.992238  f 
me 


=790  12'  20". 


2.337438 
1.293268 


a  sin.  (u  -f  v) 
mV 


a2  sin.2  ( 
2mc 
sin.  u 

2mc  sin.  u 
sin.  v  sin 
2mc  sin.  u 


11.07 


=  122.55 


log. 


log    - 


sin.  v  sin.  (u  +  v) 


=  161.73 


1.044164 
__  2 

log.     2.088328 

2.638468 
-1.698970 

2.337458 
—  1.871339 

log.     2.208797 


Whence,  x= 11.07  ±1/284.28  =  27.96  or  —5.82. 
Here  ^6^=27.96.     Having  AP,  AG,  and  the  angle  PAG,  we 
can  compute  the  angle  APG. 


DIVISION    OF    LANDS.  131 

PROBLEM  II. 

To  divide  a  triangle  into  THREE  PARTS  having  the  ratio  of  the  three 
numbers  m,  n,  p. 

CASE  1.  By  lines  drawn  from  one  angle  of  the  triangle  to  the 
opposite  side. 

Let  ADE  be  the  triangle  and  A  the 
angle  from  which  the  lines  are  to  be 
drawn. 

Divide  DE  the  opposite  side  into 
parts  in  the  ratio  of  m,  n,  and  p,  and 
from  the  points  of  division   0  and  Bt 
draw  A  C,  AJB,  and  the  triangle  is  divi- 
ded as  required. 

Demonstration. — The  areas  of  triangles  a  e  as  their  bases  multiplied 
into  their  altitudes,  but  here  all  the  triangles  have  the  same  altitude  ; 
therefore  multiplying  the  bases  into  that  altitude  gives  the  same 
proportional  product,  and  the  areas  of  the  triangles  are  as  m,  n,  p. 

CASE  2.  By  lines  parallel  to  one  of  the  sides. 

Let  ABC  be  the  triangle.  Divide  its 
numerical  area  into  three  parts  in  the  ratio 
of  m,  n,  p.  Conceive  the  problem  solved 
and  DF,  EG,  the  division  lines  parallel 
to  AB.  We  are  to  determine  the  numerical 
values  of  CD,  and  CE.  CB  is  known, 
put  it  equal  to  a.  Put  CD=x.  CE=y. 

Now   as   similar  triangles   are   to   one 
another  as  the  squares  of  their  homologous  sides,  therefore 
x2  :  a2  :  :  m  :  m-\-n-{-p. 


Whence, 
In  the  same  manner, 


In  this  manner  we  might  divide  the  triangle  into  any  proposed 
number  of  parts,  having  given  ratios. 

CASE  3.  By  lines  drawn  from  a  given  point  on  one  of  the  sides  of 
the  triangle. 


132  SURVEYING. 

Let  ABC\)Q  the  given  triangle,  and  P 
the  grven  point  on  the  side  AB. 

It  is  required  to  draw  lines  from  P,  as 
PD  and  PE,  dividing  the  triangle  into 
three  parts  me,  nc,  pc,  that  is,  assume  the 
given  numerical  area  to  be  (mc-{-nc-\-pc), 
then  the  required  parts  will  be  me,  nc,  and  pc^* Put  AD=x,  then 
(by  Prob.  Ill,  Mens.) 

•J  ax  sin.  A=mc  or  *=    ^mc 
a  sin.  A 

By  comparison,  y=._2^L. 

b  sin.  B 

When  me  and  pc  are  cut  off,  nc  is  left.  Having  a  and  x,  the  angle 
A  DP  is  easily  determined. 

In  a  similar  manner  we  can  divide  the  triangle  into  any  proposed 
number  of  parts,  by  lines  drawn  from  the  given  point  P. 
CASE  4.  By  lines  drawn  from  a  given  point  within  the  triangle. 

Let  ABC  be  the  given  triangle,  and  P 
the  given  point  within  it. 

A  variety  of  lines  may  be  drawn  from 
P,  to  divide  the  triangle  into  the  parts 
required.  Conceive  PD,  PF,  and  PE 
to  make  the  requisite  division. 

As  P  is  a  given  point,  AP,  PB,  and  PC  are  known  lines,  and 
the  angles  DAP,  PAE  are  known  angles. 

Take  AD=b,  any  convenient  assumed  value.  Take  AE=x. 
Put  AP=a.  Then, 

\al  sin  ZMP=area  A  DAP 

Also,  \ax  sin  PAE—  area  A  PAE 

*  Suppose  we  had  a  triangular  piece  of  ground  containing  320  square  rods, 
and  we  wished  to  divide  it  into  three  parts  in  the  ratio  of  2,  3,  and  5,  what  is 
the  area  of  each  of  the  parts  ? 

We  decide  it  thus  :  w=2.     n=3.    p=5.    c  is  at  present  unknown,  but, 
roe-{-nc-{-pc=320 

That  is,  10c=320  or  c=32. 

Whence,  mc=64.    nc=96.    pc=160. 

The  quantity  c  becomes  known  on  dividing  the  area,  and  the  parts  separately 
me,  nc,  and  pc,  are  always  known. 


DIVISION   OF   LANDS.  133 

Conceive  the  triangle  ABC,  divided  into  three  parts,  in  the  ratio 
of  m,  n,  p  ;  and  conceive  ADPE  to  be  one  of  these  parts  repre- 
sented by  me.  Then 

ab  sin.  DAP+ax  sin.  PAE—^mc 

2mc — ab  sin.  DAP 
Whenoe'          *=     am*.  PAS 

Had  we  taken  I  greater  than  we  did,  a;  would  have  been  less,  and 
a  variety  of  lines  could  be  drawn  as  well  as  PD  and  PE,  and  the 
same  area  cut  off. 

Having  x,  we  have  EB  as  a  known  quantity,  and  by  the  two 
triangles,  PEB  and  BPF,  we  determine  y  in  precisely  the  manner 
as  we  found  x  :  thus,  we  have  two  parts  of  the  triangle  me  and  pcf 
and,  consequently,  the  remainder  DPFQ  corresponds  to  nc. 

CASE  5.  By  lines  drawn  from  a  given  point  without  the  triangle, 

Let  AB  C  be  the  triangle,  and 
P  the  given  point  without  it. 

Divide  the  numerical  area  of 
the  triangle  into  the  three  re- 
quired parts,  me,  nc,  and  pc,  as 
in  former  cases.  Draw  PD, 
cutting  off  the  portion  me,  as 
in  Case  6  of  the  last  problem; 
then  cut  off  the  two  portions  (  mc-^-nc)  by  the  line  PO:  and  the 
portion  pc  will  be  left.  Or,  we  may  cut  off  pc,  and  the  portion  nc 
will  be  left. 

PROBLEM  III. 

To  divide  a  triangle  into  three  parts,  having  the  ratio  of  m,  n,  and 
p,  by  three  lines  drawn  from  the  three  angular  points  to  some  point 
within. 

Divide  any  side,  as  A  C,  into  three 
parts  in  the  proportion  of  m,  n, 
and  p. 

Let  Aa  represent  the  portion  cor- 
responding to  m,  and  Cc  the  part  cor- 
responding to  p. 

Through  a,  draw  ab  parallel  to  AB, 


134  SURVEYING. 

and  through  c,  draw  cd  parallel  to  CB.  Where  these  two  lines  in- 
tersect is  P9  and  the  triangle  ABC  is  divided  into  three  triangles, 
APB,  CP£,anAAPC. 

Demonstration. —  Any  triangle  having  AB  for  its  base,  and  its 
vertex  in  the  line  ab,  will  have  the  same  ratio  to  the  triangle  ABC, 
as  Aa  has  to  A  C,  that  is,  as  m  to  m-\-n-\-p. 

Also,  the  triangle  CPB  is  to  ABC,  as  Cc  is  to  CA,  that  is,  asp  to 
m+n+p,  for  triangles  on  the  same  base  are  to  one  another  as  their 
altitudes.  If  these  two  triangles,  APB  and  CPB,  are  in  due  pro- 
portion, the  third  one,  AP  C,  is  in  due  proportion,  of  course. 

PROBLEM  IV . 

To  divide  a  quadrilateral  into  two  parts,  having  any  given  ratio 
m  to  n. 

CASE  1.  By  a  line  drawn  from  a  given  point  in  the  perimeter. 

Let  ABCD  be  the  given  quadri- 
lateral, and  P  the  given  point  in  the 
side  AB. 

It  is  required  to  determine  the 
magnitude,  and  the  direction  of  the  line  PG,  which  divides  the 
figure  into  parts  in  the  ratio  of  m  to  n.  All  the  sides  and  angles 
of  the  quadrilateral  are  known,  and  its  area  is  known.  AB  and 
D  C  are,  or  are  not  parallel ;  if  they  are  parallel  the  figure  is  a  tra- 
pezoid,  then  the  method  of  finding  G,  in  the  opposite  side,  is 
easy  and  obvious.  If  AB  and  CD  are  not  parallel,  we  can  pro- 
duce them  and  form  a  triangle  in  the  one  direction  or  the  other ;  by 
this  figure  we  form  the  triangle  B  CE,  whose  area  we  may  repre- 
sent by  t. 

As  BC,  and  the  angles  as  B  and  Care  all  known,  the  triangle 
EBQ  is  determined  in  all  respects. 

As  PB  is  known,  PE  is  known,  and  designate  EG  by  x.  Let 
cm  and  en  designate  the  portions  of  the  quadrilateral  after  it  is  di- 
vided, and  let  cm  represent  the  part  BPGC. 

Put  PE=a. 

Now  (  by  Problem  III,  Mensuration  ),  we  have 
\ax  sin.  JE=t+  cm. 


DIVISION    OF    LANDS.  135 


We  now  have  the  numerical  value  of  x,  from  which  we  subtract 
EC,  and  we  have  CG,  which  being  measured  from  C  will  give  the 
point  G,  through  which  to  draw  the  line  from  P,  to  divide  the 
figure  as  required. 

CASE  2.  By  a  line  making  a  given  angle  with  one  of  the  sides. 

If  the  division  line  makes  a  given  angle  with  one  side,  it  must 
also  make  a  known  angle  with  the  opposite  side. 

Taking  the  last  figure,  conceiving  PG  to  take  a  given  direction 
across  AB  and  CD,  so  as  to  cut  off  the  area  me. 

As  in  the  former  case,  let  t  represent  the  area  of  the  triangle  EEC, 
to  this  add  me,  and  we  have  the  area  of  the  triangle  P  GE.  But  in 
this  case  P  is  not  a  given  point,  and  EP  is  not  known. 

Put  JSG=x.  Let  P  represent  the  given  angle  at  P,  and  G  the 
given  angle  at  G.  Now,  by  trigonometry, 

sin.  P  :  x  :  :  sin.  G  :  EP 


Or, 

sin.  P 

(Prob.  III.  Mens.)   Sin'  S  sm'  <*x*  =  t+mc 
2sin.P 


Whence, 


From  x  we  take  EC,  measure  off  the  remainder  along  CD,  to  the 
point  G,  there  making  the  given  angle,  and  the  figure  will  be  di- 
vided as  required. 

CASE  3.  By  a  line  drawn  through  a  given  point  within  the  quad- 
rilateral. 

Let  ABCD  be  the  quadrilate- 
ral as  before,  and  P  the  given 
point  within  it  ;  and  as  P  is  the 
given  point,  EP  is  a  known  line, 
and  the  angles  PEE,  PEG  are  known. 

Let  t  equal  the  area  of  the  triangle  EBCt  as  before,  and  me  the 
area  GHCB.  Put  JEff=x,  and  EG=,y. 


136  SURVEYING. 

Now  we  have  a  problem  precisely  like  Case  4,  Problem  I,  of  this 
chapter ;  therefore,  further  explanations  would  be  superfluous. 

CASK  4.  By  a  line  drawn  through  a  given  point  without  the 
quadrilateral. 

Let  ABCDte  the  quad- 
rilateral as  before,  and  P  the 
given  point  without  it. 

By  producing  the  two 
sides  AB  CD,  we  form  the 
triangle  ADE.  Let  the  area  of  the  triangle  B  CE  be  represented 
by  t,  and  the  part  GHCB  by  me,  then  from  a  given  point  P,  with- 
out a  triangle,  we  are  required  to  draw  a  line  Pff,  to  divide  the 
triangle  into  two  given  parts,  and  this  is  Case  6,  of  Problem  I  of  this 
chapter,  which  has  been  fully  investigated. 

REMARK.  —  By  extending  the  principles  of  these  several  cases  we  may  divide 
a  quadrilateral  into  three  or  more  parts. 

PROBLEM  V. 

To  divide  any  polygon  (regular  or  irregular)  into  two  parts  having 
a  given  ratio,  m  ton,  by  a  line  drawn  through  a  given  point. 

CASE  1 .  When  the  given  point  is  on  one  of  the  sides  of  the  polygon. 

Let  ABODEF  be  the  polygon,  and 
(mc-\-nc)  express  its  numerical  area.  Let  P 
be  the  given  point  on  the  side  AB.  Let  the 
surveyor  run  a  random  line  as  near  to  the 
line  required  as  his  judgment  permits,  and 
generally  it  will  be  best  to  run  a  line  from 
P  to  one  of  the  opposite  angular  points. 

In  this  figure,  let  PE  represent  such  a  random  line,  and  let  the 
surveyor  compute  the  area  of  the  figure  PAFEt  thus  cut  off,  which 
area  will  be  equal  to,  or  greater,  or  less  than  one  of  the  required 
parts.  We  will  suppose  it  less  ;  then  subtract  it  from  the  required 
portion  me,  and  let  the  triangle  PEG  represent  that  known  difference, 
which  we  shall  designate  by  t. 

PE  is  known,  the  angle  PEG  is  known  ;  and  put 
Then,  %PE  x  sin.  PEG=t 

Whence,  x= 

P^sin.  PEG 


DIVISION   OF  LANDS.  137 

This  determines  the  point  G,  and  PG  divides  the  polygon  as 
required. 

CASE  2.  When  the  given  point  is  within  the  polygon. 

Let  ABCDEFte  the  polygon  as  before,  and  (mc-\-nc)  express 
its  numerical  area  ;  also,  let  P  be  the  given  point  within.  Through 
P  let  the  surveyor  run  the  random 
line,  HPK,  measuring  from  #to  P, 
and  from  P  to  K,  let  him  also 
observe  the  angles  that  this  line 
makes  with  the  sides  of  the  polygon 
AFtmd  CD,  and  compute  the  area 
HABCK,  and  note  the  difference 
between  it  and  mcy  the  required 
portion  of  the  polygon  ;  call  this 
difference  d,  a  known  quantity. 

Let  hPk  represent  the  true  line  through  P,  which  divides  the 
polygon  as  required  ;  but  this  line  diminishes  the  area  HABCK,  by 
the  triangle  PKk,  and  increases  it  by  the  triangle  PHh. 

Therefore  the  difference  of  these  triangles  must  equal  d. 

The  sine  of  the  angle  AHP,  has  the  same  numerical  value  as  the 
sine  of  PHh,  and  the  sine  of  the  angle  PKD  has  the  same  numeri- 
cal value  as  the  sine  of  the  angle  PKk. 

Put  the  angle  AlfP—u,  and  the  angle  PJ£D—v;  let  the  acute 
f  article  angles  at  P  be  designated  by  the  letter  P. 

Let  ffP=a,  PK=b,  hP=x,  Pk=y 

In  the  triangle  PhH,  we  have 

sin.  u  :  x  :  :  sin.  (u  —  P)  :  a 

a  sin.  u  /  <  \ 

*~ 


The  area  of  the  triangle  PhH=^ax  sin.  P 
Also,  PkK=\by  sin.  P 

Whence,  b  sin.  Py—  a  sin.  Px=Zd  (3) 

By  substituting  the  values  of  x  and  y,  taken  from  (1)  and  (2), 
we  have, 


138  SURVEYING. 

sm.  P  sin.  sm.  ?  sin. 


sin.  (y-P)          sin.  («—  P) 

Equation  (4)  contains  only  one  unknown  quantity  P,  the  value  of 
P,  or  the  angle  HPh  can  therefore  be  deduced. 
£2  /  sin.  P  sin,  ti  \     ^  /  sin.  P  sin,  u  \ 

Vsin.  #  cos.  P—  cos.  v  sin.  P/         \sin.  u  cos.  P—  cos.  #  sin.  P/ 
=2rf. 

Dividing  the  numerator  and  denominator  of  the  first  fraction  by 
(sin.  P  sin.  v),  and  of  the  second  fraction  by  (sin.  P  sin.  u.),  recol- 
lecting that  cosine  divided  by  sine  gives  cotangent.  Thus  we  shall 

°btain 


,cot.  P — cot.  v/  \cot.  P — cot.  u. 
This  last  equation  shows  the  surveyor  that  if  he  can  make  it  conven- 
ient to  run  his  random  line  from  P,  perpendicular  to  one  of  the  sides, 
his  equation  will  be  less  complex.  For  instance,  if  AIIP=90°,  its 
cotangent  will  be  0,  and  cot.  u  would  then  =0,  and  equation  (5) 
would  become. 

b2  a2  (6) 

cot.  P — cot.  v~~cot.  P~~ 
For  the  sake  of  convenience  put  cot.  P=z,  and  cot.  v—c. 

Then 

b2  a2 


Or,  -•  '  '  f'J— b 


The  numerical  value  of  z  will  be  the  numerical  value  of  cot.  P ;  its 
logarithm  taken,  and  10  added  to  the  index  will  be  logarithmic  cot. 
in  our  table.  The  same  remarks  will  apply  to  cot.  v  or  c. 

CASE  3.  When  the  given  point  is  without  the 
polygon. 

Let  ABCDE  be  the  polygon,  and  P  the 
given  point  without  it. 

From  the  last  case  we  learn  that  the  surveyor 
had  better  run  his  random  line  perpendicular 
to  one  of  the  sides,  therefore  let  PHO  be  the 
random  line,  perpendicular  to  AE. 


DIVISION   OF   LANDS.  139 

As  before,  compute  the  area,  AE&Hy  subtract  it  from  me,  the  dif- 
ference is  the  difference  between  the  triangles  POL  and  PHK. 
Draw  PKL  the  line  that  divides  the  polygon  as  required. 

Put  PH=a,  PG=b,  PK^x,  PL=y,  angle  #=90°,  angle  PQE 
*=su,  and  the  angle  at  P,  designated  by  P. 

The  triangle    PHK-=\ax  sin.  P 
PGL—\ln)  sin.  P 

Whence,  b  sin.  Py  —  a  sin.  P  z=2d.  (1) 

Here  (2)  represents  a  similiar  quantity  as  in  the  last  case. 
In  the  triangle  PHIZ,  we  have 

t  :  x  :  :  cos.  Pa,  or  ar=  —  ^-_  /Qx 

cos.  P  (*) 

In  PGL,  sin.  u  :  y  :  :  sin.  («—  P)  :  b. 

b  sin.  u 


When  the  values  of  #  and  y  are  substituted  in  (1)  we  have 

sin.  P  sin.  u        sin.  P 
•    *%in.  (M-JP)-aac-^>=2d  (4) 

(sin.  P  sin.  u  \         sin.  P 

sin.  M  cos.  P—cos.  u  sin.  p)~~a2^TP=2d  (5) 

b2  __    qa 

Or>  cot.  P—  cot.M^oTTP^  (6) 

This  equation  is  exactly  similar  to  equation  (6)  of  the  last  case, 
and  it  is  reduced  in  the  same  manner. 


PROBLEM  VI. 

To  divide  a  polygon  into  three  or  more  parts,  having  a  given  ratio, 
m,  ntp,  q,  by  lines  passing  through  a  given  point. 

This  problem  admits  of  three  cases. 

CASE  1.  When  the  given  point  is 
on  one  side  of  the  polygon. 

Divide  the  numerical  area  of  the 
whole  into  parts,  me,  nc,  pc,  qc,  cor- 
responding to  the  given  ratio.  Unite 
these  into  two  parts  (mc-\-nc)  and 
(pc+qc). 


140  SURVEYING. 

From  the  given  point  P,  draw  PG,  by  Case  1,  Problem  V, 
so  as  to  divide  the  polygon  into  the  two  parts  (mc-\-nc)  and  (pc-{-qc). 

We  have  now  to  divide  the  polygon  PAEG  into  two  parts,  mct 
nc,  by  the  line  PL,  and  the  polygon  PBCDG  into  two  parts,  pc 
and  qc,  by  the  line  PH. 

CASK  2.  When  the  given  point  is  within  the  polygon. 

Let  ABCDE  be  the  given  poly- 
gon and  P  the  given  point  within. 

Draw  Kk  through  P,  by  Case  2, 
Problem  V,  so  that  the  area  AhkDE 
shall  equal  me,  and  the  area  JikCE 
shall  equal  (nc-\-pc),  when  the  whole 
is  required  to  be  divided  into  three 
parts  in  the  ratio  of  m,  n,  p. 

When  the  whole  is  to  be  divided  into  four  parts,  in  the  ratio  of  m, 
n,  p,  q,  then  draw  hkt  so  that  one  portion  shall  be  (mc+nc)  and 
the  other  (pc-\-qc). 

Then  we  have  P  as  a  given  point,  in  one  side  of  the  polygon, 
AlikDE,  to  divide  it  into  two  parts,  in  the  ratio  m  to  n,  and  P  a  given 
point  on  one  side  of  the  polygon,  Tik  CB,  to  divide  it  into  two  parts, 
in  the  ratio  of  p  to  q,  and  this  is  done  by  Case  1 ,  Problem  V. 

CASE  3.  When  the  given  point  is  without  the  polygon. 

Let  ABCDE F  be  the  given  poly- 
gon, and  P  the  given  point  without  it. 

Divide  the  numeral  area  into  the 
required  proportional  parts,  me,  nc,pc, 
<fec.,  as  many  as  required. 

From  the  point  P  draw  the  line  PH, 
as  directed  in  Case  3,  Problem  V,  di- 
viding the  polygon  into  two  parts,  me 
and  (nc+^c-j-(fec.). 

Then  divide  the  polygon,  GHED  CB,  into  two  parts,  one  of  which 
is  nc,  and  the  other  (pc-\-qc,  <fec.),  and  thus  we  can  proceed  and 
cut  off  one  portion  after  another,  as  many  as  may  be  required. 

The  application  of  the  foregoing  principles  will  meet  any  case  that 


DIVISION   OF    LANDS.  141 

can  occur  in  the  division  of  lands  ;  and  we  now  close  this  subject 
with  the  following  practical 

EXAMPLES . 

1.  A  triangular  field,  whose  sides  are  20,  18,  and  16  chains,  is  to 
have  a  piece  of  4  acres  in  content  fenced  off  from  it,  by  a  right  line 
drawn  from  the  most  obtuse  angle  to  the  opposite  side.     Required  the 
length  of  the  dividing  line,  and  its  distance  from  either  extremity  of  the 
line  on  which  it  falls  ? 

Ans.  Length  of  the  dividing  line,  13  chains,  89  links,  if  run 
nearest  the  side  16.  Distance  it  strikes  the  base  from  the  next  most 
obtuse  angle  is  5.85  chains. 

2.  The  three  sides  of  a  triangle  are  5,  12,  and  13.     If  two-thirds 
of  this  triangle  be  cut  off  by  a  line  drawn  parallel  to  the  longest  side,  it 
is  required  to  find  the  length  of  the  dividing  line,  and  the  distance  of 
its  two  extremities  from  the  extremities  of  the  longest  side. 

Ans.  Distance  from  the  extremity  on  5,  is  5(«/3— ^2);  on  the 
side  of  12,  it  is  12(«/3— ^2)  ;  both  divided  by  ^1. 
The  division  line  is  13^/f. 

3.  It  is  required  to  find  the  length  and  position  of  the  shortest  possible 
line,  which  shall  divide,  into  two  equal  parts,  a  triangle  whose  sides  are 
25,  24,  and  7  respectively. 

REMARK.  —  It  is  obvious  that  the  division  line  must  cut  the  sides  25  and  24, 
and  to  make  it  the  shortest  line  possible,  the  triangle  cut  off  must  be  Isosceles. 

Ans.  The  division  line  makes  an  angle  with  the  sides  25  and  24 
of  81°  52'  1 17',  and  its  length  is  4.899. 

4.  The  sides  of  a  triangle  are  6,  8,  and  10.   It  is  required  to  cut 
off  nine-sixteenths  of  it,  by  a  line  that  shall  pass  through  the  center  of 
its  inscribed  circle. 

Ans.  The  division  line  cuts  the  side  of  10,  at  the  distance  of  7.5 
from  the  most  acute  angle,  and  on  the  side  of  8,  at  the  distance  6 
from  the  most  acute  angle. 

5.  Two  sides  of  a  triangle,  which  include  an  angle  of  70°,  are  14 
and  17  respectively.     It  is  required  to  divide  it  into  three  equal  parts, 
by  lines  drawn  parallel  to  its  longest  side. 


142  SURVEYING. 

Ans.  The  first  division  line  on  the  side  17,  cuts  that  side  at  the 
distance—  —  ;  the  second  division  line  —  ^-.     The  side  14  is  cut  at 


-Hand 


6.  Three  sides  of  a  triangle  are  1751,  1257.5,  and  2364.5.     The 
most  acute  angle  is  31°  17'  19".     This  triangle  is  to  be  divided  into 
three  equal  parts  by  lines  drawn  from  the  angular  points  to  some  point 
within.     Required  the  lengths  of  these  lines. 

Ans.  The  line  from  the  most  acute  angle  is  1322.42,  and  from  the 
next  most  acute  angle  1119 

7.  The  legs  of  a  right-angled  triangle  are  28  and  45.     Required  the 
lengths  of  lines  drawn  from  the  middle  of  the  hypotenuse,  to  divide  it 
into  four  equal  parts. 

Ans.  A  line  drawn  from  the  middle  of  the  hypotenuse  to  the 
right  angle,  divides  the  triangle  into  two  equal  parts. 

8.  In  the  last  example,  suppose  the  given  point  on  the  hypotenuse  at 
the  distance  of  13  from  the  most  acute  angle,  whereabouts  on  the  other 
sides  will  the  division  lines  fall  to  divide  the  triangle  into  three  equal 
parts  ? 

N.  B.  The  sine  of  an  acute  angle  to  any  right-angled  triangle  is 
equal  to  the  side  opposite  that  angle  divided  by  the  hypotenuse. 

Ans.  Both  division  lines  fall  on  the  side  28,  distance  of  the  first 
from  the  acute  angle  12^-,  of  the  second  24f  a. 

9.  There  is  a  farm  containing  64  acres,  commencing  at  its  south 
westerly  corner,  the  first  course  is  North  15°  E.,  distance  12  chains  ; 
the  second  is  N.  80°  E.  (distance  lost),  the  third  S.  (distance  lost), 
the  fourth  is  N.  82°  W.  (distance  lost),  to  the  place  of  beginning.     It 
is  required  to  determine  the  distances  lost. 

OBSERVATION.  —  Extend  the  northern  and  southern  boundary  westward,  and 
thus  form  a  triangle  on  the  west  side  of  12. 

Ans.  The  2nd  side  is  35.816  ch.     3rd,  23.21  ch.     4th,  38.76  ch. 

The  two  following  problems  are  from  GUMMERE'S  Surveying,  and 
are  considered  very  difficult. 


DIVISION    OF    LANDS.  143 

1.  There  is  a  piece  of  land  bounded  as  follows  : 
Beginning  at  the  south-west  corner  ;  thence, 

1.  JV.  14°  00'  W.y  Distance  15.20  chains=a; 

2.  N.  70°  30'  E.y         "        20.43      "    =6  ; 

3.  S.    6°  00'  E.y        "        22.79      "    =c; 

4.  N.  86°  30'  W.9        "        18.00      "    =d. 

Within  this  lot  there  is  a  spring  ;  the  course  to  it  from  the  second 
corner  is  S.  75°  E.t  distance  7.90  chains.  It  is  required  to  cut  of  ten 
acres  from  the  west  side  of  this  lot,  by  a  line  running  through  the  spring. 
Where  will  this  line  meet  the  fourth  sidet  that  is,  how  far  from  the 
first  corner  ?  Ans.  4.6357  chains. 

First  make  a  plot  of  the  field.     It  is  as  here  represented. 

Produce  the  sides  b  and  dt  the  second 
and  fourth,  until  they  meet  at  O.  Let  S 
be  the  position  of  the  spring,  and  join  SO. 
We  may  or  may  not  find  the  contents  of 
the  field*  :  it  is  not  necessary  for  the 
location  of  the  line  LSH. 

It  is  necessary  to  find  the  area  of  the  triangle  AE  O.  Conceive 
a  meridian  line  run  through  B  ;  then  we  perceive  that  the  angle 
ABO=\  4°+  70°  30'  =84°  30'.  Conceive  also  a  meridian  line  to 
be  run  through  A,  and  then  we  perceive  that  the  angle  BA6r=Q6° 
30'—  14°=72°  30'  ;  whence  .4  #.5=  23°.  With  the  angles  and  the 
side  AB=  15.20,  we  readily  find  ^£=38.72,  £#=37.10,  and  the 
area  -4(jj5=280.65  square  chains. 

It  is  necessary  to  find  the  line  OS  and  the  angle  BOS.  From 
the  given  direction  of  the  lines  BO  and  BS,  we  find  the  angle 
GBS=  145°  30';  and  then  from  the  triangle  OBS,  we  find 
BOS=5°  51'  30",  and  #£=43.83.  Also  we  have  the  angle 
=  17°  8'  30'. 


To  the  area  of  the  triangle  ^#5=280.65,  add  10  acres,  or  100 
square  chains  :  then  the  area  of  the  triangle  OLffmust  equal  380.65 

*  When  we  have  four  sides  only,  and  all  the  angles,  as  in  this  field,  the  best 
method  of  finding  the  contents  is  by  conceiving  it  to  be  two  triangles.  Thus  in 
this  case  the  area  is  represented  by 

£  ab  sin.  ABC+l  dc  sin.  CD  A. 


144  SURVEYING. 

square  chains  ;  but  OL  and  Off  are  both  unknown.     Put  GL=y, 
GH=x  :  then  we  shall  have  the  equation. 

xy  sin.  23°=2(380.65).  (1) 

It  is  obvious  that  the  sum  of  the  two  triangles  LQS,  SGH  is 
equal  to  the  triangle  QLH. 

But    #£=m=43.S3,   sin.  23°=P,   sin.  (17°  8'  30")=  Q,   sin. 
(5°  51'  30")=JR,  and  2(380.65)  or  761.3=a  :  then  we  have 

Pxy—ay  ( I ) 

and  •     Rmy-\-  Qmx=a,  (2) 

From  (1),  y=JL.  This  value  put  in  (2),  gives 


(3) 

JTX 

whence,  **— JL#  =  — —.  (4) 

We  now  find  the  numerical  values  of and  — -  by  logarithms, 

as  follows  : 

As  our  radius  is  unity,  we  dimmish  the  indices  of  the  logarithmic 
sines  by  10. 

log.  a,  2.881556  log.  a,      2.881556 

log.  m,     1.641771  log.  /?,— 1.008880 

log.  £,—1.469437 


1.890436 


1.111208     1.111208 


58.932     1 .770348     log.  P,— 1 .59 1 878 
log.  £,—1.469437 


—1.061315        —1.061315 


674.72         -  ...          2.829121 

Equation  (4)  now  becomes 

x2— 58.932*=— 674.7  ; 

whence  x=  OH=  43.366  :  from  which  subtract  (£4=38.72,  and 
we  have  AH  the  distance  required  =4.646,  which  differs  from  the 
given  answer  about  one  link  of  the  chain. 


DIVISION   OF    LANDS.  J45 

LEMMA.  Find  the  point  in  any  trapezoid,  through  which  any 
straight  line  which  meets  the  parallel  sides  will  divide  the  trapezoid  into 
two  equal  parts. 

Let  ABCD  be  the  trapezoid. 

Bisect  the  parallel  sides  AE  and  CD  in 
the  points  n  and  m.  Join  mn,  and  bisect 
mn  in  0,  and  0  is  the  point  required. 

Any  line  meeting  the  parallel  sides,  and  passing  through  0,  will 
divide  the  trapezoid  into  two  equal  trapezoids.  It  is  obvious  that 
the  line  mn  divides  the  figure  into  two  equal  parts,  because  the  sum 
of  the  parallel  sides  is  the  same  in  each.  Now  draw  any  other  line 
through  0,  as  p  Oq  :  the  trapezoid  pqBD—mnBD ;  because  the 
triangle  Omp=qOn,  and  one  triangle  is  cut  off  and  the  other  is  put 
on  at  the  same  time.  The  triangles  are  equal,  because  mO=  On, 
the  angle  pm  0—  Onq,  and  the  opposite  angles  at  0  are  equal : 
therefore  pm=nq ;  and  whatever  more  than  Cm  is  taken  on  one 
side,  the  equal  quantity  qn  less  than  An  is  taken  on  the  other  side. 

Another  method  of  finding  the  point  0,  is  to  bisect  A  0  in  IT,  and 
draw  HO  parallel  to  AE  or  CD,  and  equal  to  one-fourth  the  sum 
of  AE  and  CD. 

2.  There  is  a  piece  of  land  bounded  as  follows  : 
Beginning  at  the  westernmost  point  of  the  field  ;  thence, 

1.  N.  35°  15'  K,    23.00  chains  ; 

2.  N.  75°  30'  E.,    30.50       " 

3.  S.    3°  15'  K,    46.49       " 

4.  N.  66°  15'  W.,  49.64       " 

M  is  required  to  divide  this  field  into  four  equal  parts,  by  two  lines, 
one  running  parallel  to  the  third  side,  the  other  cutting  the  first  and 
third  sides.  Find  the  distance  of  the  parallel  line  from  the  first  corner 
measured  on  the  fourth  side,  and  the  bearing  of  the  other  line. 

Ans.  Distance  to  the  parallel,  32.60  chains ;  Bearing  of  the  other 
side,  AS.  88°  22'  E. 
10 


146  SURVEYING. 

CHAP  TER     VII. 

TRIANGULAR    SURVEYING:     THE    PLANE 

TABLE  — ITS     DESCRIPTION     AND     USES: 

MAPPING:     MARINE     SURVEYING. 

TRIANGULAR  SURVEYING,  as  here  understood,  requires  the  actual 
measurement  of  only  one  line,  and  all  other  lines  can  be  deduced 
from  this  by  means  of  observed  angles  forming  triangles,  of  which 
this  measured  line  forms  a  base  of  the  first  triangle  in  the  series  or 
chain  of  triangles. 

Some  of  the  other  lines  however  should  be  measured  after  being 
computed,  as  a  test  to  the  accuracy  or  inaccuracy  of  the  operations. 

Let  AE  represent  a  base  line 
which  must  be  very  accurately 
measured,  for  any  error  on  AE  will 
cause  a  proportional  error  in  every 
other  line. 

If  at  A  we  measure  the  angles 
EA  C,  BAD,  and  at  E  we  measure 
or  observe  the  angles  ABC,AED, 
we  then  have  sufficient  data  to  deter- 
mine the  points  C  and  D,  and  the  line  CD. 

With  equal  facility  that  we  determine  the  point  C,  we  can  deter- 
mine the  point  E,  or  F,  or  G,  or  any  other  visible  point. 

Thus  we  may  determine  all  the  sides  and  angles  of  the  figure 
CEFGHD,  or  any  visible  part  of  it,  by  triangulating  from  the 
base  AE. 

The  lines  forming  the  triangles  are  not  drawn,  except  those  to  the 
points  G  and  D  ;  we  omitted  to  draw  others  to  avoid  cor/fusion. 

After  any  line,  as  FG,  has  been  computed,  it  is  well  to  measure  it, 
and  if  the  measurement  corresponds  .with  computation,  or  nearly  so, 
we  may  have  full  confidence  of  the  accuracy  of  the  work  as  far  as  it 
has  been  carried. 

We  may  take  CD  as  the  base,  and  determine  any  visible  number 
of  points,  as  A,  E,  Hy  Fy  G,  <fec.,  trace  any  figure  and  determine  its 
area,  or  show  the  relative  positions  and  distances  of  objects  from 
each  other,  such  as  buildings,  monuments,  trees,  &c. 


THE  PLANE  TABLE.  147 

But  to  make  the  computation,  triangle  after  triangle,  for  the  sake 
of  making  a  map,  would  be  very  tedious,  and  to  measure  every  side 
and  angle  would  be  as  tedious,  and  to  facilitate  this  land  of  operation 
we  may  have  an  instrument  called  the 

PLANE      TABLE. 

The  plane  table  is  exactly  what  the  name  indicates  ;  it  is  a  plane 
board  table,  about  two  feet  long,  and  twenty  inches  wide,  resting  on 
a  tripod,  to  which  it  is  firmly  screwed,  yet  capable  of  an  easy  motion 
on  its  center,  having  a  ball  and  socket  like  a  compass  staff. 

Directly  under  the  table  is  a  brass  plate,  in  which  four  milled 
screws  are  worked,  for  the  purpose  of  adjusting  the  table,  the 
screws  pressing  against  the  table. 

To  level  the  table,  a  small  detached  spirit  level  may  be  used. 
The  level  being  placed  on  the  table  over  two  of  the  screws,  the 
screws  are  turned  contrary  ways,  until  the  level  is  horizontal ;  after 
which  it  is  placed  over  the  other  two  screws,  and  made  horizontal  in 
the  same  manner. 

The  table  has  a  clamp  screw,  to  hold  it  firmly  during  observa- 
tions, and  also  a  tangent  screw,  to  turn  it  minutely  and  gently,  after 
the  manner  of  the  theodolite. 

The  upper  side  of  the  table  is  bordered  by  four  brass  plates, 
about  an  inch  wide,  and  the- center  of  the  table  is  marked  by  a  pin. 

About  this  center,  and  tangent  to  the  corners  of  the  table,  conceive 
a  circle  to  be  described.  Suppose  the  circumference  of  this  circle  to 
be  divided  into  degrees  and  parts  of  a  degree,  and  radii  to  be  drawn 
through  the  center,  and  each  point  of  division. 

The  points  hi  which  these  radii  intersect  the  outer  edge  of  the 
brass  border,  are  marked  by  lines  on  the  brass  plates ;  these  lines 
of  course  show  degrees  and  parts  of  degrees ;  they  are  marked  from 
right  to  left,  from  0  to  180°  on  both  sides,  but  on  some  tables  the 
numbers  run  all  the  way  round,  from  0  to  360°. 

Near  the  two  ends  of  the  table  are  two  grooves,  into  which  are 
fitted  brass  plates,  which  are  drawn  down  into  their  places  by  screws 
coming  up  from  the  under  side.  The  object  of  these  grooves  and 
corresponding  plates,  is  to  hold  down  paper  firmly  and  closely  to 
the  table. 


148  SURVEYING. 

The  paper  before  being  put  on,  should  be  moistened  to  expand  it, 
then  carefully  drawn  over  the  table,  and  fastened  down  by  the  plates 
that  fit  into  the  grooves ;  on  drying,  it  will  fit  closely  to  the  table. 

A  delicate  fine  edged  ruler  is  used  with  the  plane  table,  it  has 
vertical  sights  ;  the  hairs  of  the  sights  are  in  the  same  vertical  plane 
as  the  edge  of  the  ruler. 

A  compass  is  sometimes  attached  to  the  table,  to  show  the  bear- 
ings of  the  lines  ;  but  the  most  practical  mathematicians  prefer  each 
instrument  by  itself. 

The  plane  table  may  be  used  to  advantage  for  three  distinct 
objects. 

1 .  For  the  measurement  of  horizontal  angles. 

2.  For  the  determination  of  the  shorter  lines  of  a  survey,  both  as 
to  extent  and  position. 

3.  For  the  purpose  of  mapping  down  localities,  harbors^  water- 
courses, &c. 

1 .   To  measure  a  horizontal  angle. 

Place  the  center  of  the  table  over  the  angular  point,  by  means 
of  a  plumb.  Level  the  table,  then  place  the  fine  edge  of  the  ruler 
against  the  small  pin  at  the  center  ;  direct  the  sights  to  one  object, 
and  note  the  degree  on  the  brass  plate  ;  then  turn  the  ruler  to  the 
other  object,  and  note  the  degree  as  before. 

The  difference  of  the  degrees  thus  noted,  is  the  angle  sought. 

If  the  ruler  passed  overO,  in  turning  from  one  object  to  the  other, 
subtract  the  larger  angle  from  180°,  and  to  the  remainder  add  the 
smaller  angle,  for  the  angle  sought. 

2.  To  determine  lines  in  extent  and  position. 

Let  CD  be  a  base  line;  having 
the  paper  on  the  table,  all  dried  and 
ready  for  use.  Place  the  table  over 
C,  so  that  the  point  on  the  table, 
where  we  wish  C  to  be  represented, 
shall  fall  directly  over  C ;  and  place 
the  position  of  the  table  so  that  CD 
shall  take  the  desired  direction  on 
the  table. 


THE    PLANE   TABLE.  149 

Now  level  the  instrument,  and  clamp  it  fast ;  it  is  then  ready  for 
use. 

Sight  to  the  other  end  of  the  base  line,  and  mark  it  along  the  fine 
edge  of  the  ruler. 

In  the  same  manner  sight  along  the  direction  of  CE,  and  mark 
that  direction  in  a  fine  lead  line,  that  can  be  easily  rubbed  out,  the 
point  E  is  somewhere  in  that  line. 

Sight  hi  the  direction  of  F,  and  mark  the  line  on  the  paper  ;  F  is 
somewhere  La  that  line.  In  this  manner  sight  to  as  many  objects  as 
desired,  as  G,  H,  B,  A,  &c. 

Now  the  base  on  the  paper,  may  be  as  long,  or  as  short  as  we 
please  ;  suppose  the  real  base  on  the  ground  to  be  1200  feet ;  this 
may  be  represented  on  the  table  by  3,  4,  5,  10,  or  12  inches,  more 
or  less  ;  suppose  we  represent  it  by  6  inches,  then  one  inch  on  the 
paper,  will  correspond  with  200  feet  on  the  ground  (horizontally). 

Take  CD  six  inches,  and  place  a  pin  at  D,  remove  the  instrument 
to  the  other  end  of  the  base,  and  place  D  of  the  table  right  over 
the  end  of  the  base,  by  the  aid  of  a  plumb,  and  give  the  table  such 
a  position  as  will  cause  CD  on  the  table,  to  correspond  with  the 
direction  of  the  base. 

Level  the  table  and  clamp  it.  Now,  if  CD  on  the  table,  does 
not  exactly  correspond  with  the  direction  of  the  base  on  the  ground, 
make  it  correspond  by  means  of  the  tangent  screw. 

Now  from  D,  by  means  of  the  ruler  and  its  sight  vanes,  draw 
lines  on  the  paper,  in  the  direction  of  the  points  E,  F,  G,  H,  B,  A,  <kc.; 
and  where  these  lines  intersect  those  from  the  other  end  of  the  base, 
to  the  same  points,  is  the  real  localities  of  those  points,  in  proportion 
to  the  base  line.  Lines  drawn  from  point  to  point,  where  these  lines 
intersect,  as  EF,  FG,  GH,  &c.,  will  determine  the  distances  from 
point  to  point,  at  the  rate  of  200  feet  to  the  inch. 

Lines  drawn  from  the  center  of  the  table,  parallel  to  FJEa,ud  FG, 
will  determine  the  angle  EFG,  in  case  the  angle  is  required.  After 
the  points  E,  F,  G,  &c.,  have  been  determined,  the  light  pencil  lines 
to  them,  from  the  ends  of  the  base,  may  be  rubbed  out,  except  those 
that  we  may  wish  to  retain. 

Here  we  perceive  the  utility  of  the  plane  table  ;  we  have  a  multitude 
of  results,  as  soon  as  the  observations  are  made. 


153  THE    PLANE    TABLE. 


The  plane  table  will  give  us  at  once,  the  relative  distances  of 
buildings  from  the  base,  and  from  each  other,  and  if  we  are  careful 
and  particular,  we  can  obtain  the  magnitudes  of  the  buildings,  as  is 
obvious  by  the  adjoining  figure. 


This  is  most  useful  to  an  officer  or  a  spy,  who  wishes  as  exact 
knowledge  of  an  enemy's  locality  as  possible.  Or  from  a  distant 
place  AB,  we  may  examine  and  measure  any  objects  whatever,  on 
the  other  side  of  a  river,  or  give  a  correct  delineation  of  the  river 
itself. 

The  plane  table  can  be  made  very  useful  to  civil  engineers,  for 
mapping  the  localities  through  which  a  canal  or  railroad  passes. 
Take,  for  example,  a  railroad  line,  ABCDE,  represented  in  the 
next  figure.  The  lines  being  all  measured  and  marked  in  distances 
of  100  or  200  feet,  the  bases  are  all  ready  where  the  line  is 
straight. 

Set  the  table  as  at  A,  and  draw  lines  to  all  objects  that  you  wish 
to  appear  on  the  map,  both  to  the  right  and  to  the  left, —  then  move 
the  instrument  to  B,  drawing  lines  to  the  same  object  from  a  corres- 
ponding base  on  the  paper,  and  also  draw  lines  to  other  objects  fur- 
ther in  advance  on  the  line  that  may  be  seen  from  another  base. 

The  intersections  of  the  lines  from  the  extremities  of  any  base  to 
to  the  same  object  will  locate  the  object. 

When  all  the  objects  are  thus  located,  both  to  the  right  and  to 
the  left,  we  pass  on  to  new  objects,  taking  care  to  keep  at  least  two 
of  the  old  objects  in  sight,  to  connect  one  new  observation  with  those 
previously  taken.  We  now  commence  a  series  of  observations  from 
a  new  base,  which  base  must  take  its  proper  relative  position  on  the 


SURVEYING. 


151 


152  SURVEYING. 

paper,  and  if  the  paper  on  the  table  is  not  large  enough,  it  must  be 
taken  off  and  new  paper  put  on,  and  two  of  the  objects  on  the  old 
paper  must  appear  on  the  new,  and  then  these  two  papers  can  be 
put  together  so  that  the  objects  which  are  on  both  papers  will  coin- 
cide, and  then  the  two  papers  will  be  the  same  as  one,  and  thus 
we  may  put  any  number  of  papers  together  and  form  as  large  a  map 
as  we  please. 

If  the  different  bases  are  not  in  the  same  direction,  the  objects 
which  are  on  two  different  papers  on  being  put  together  will  show 
it,  and  several  papers  put  together  may  make  a  very  inconvenient 
figure  ;  but  they  must  be  put  together  and  then  a  square  sheet  of 
tissue  paper  put  over  the  whole,  and  the  map  taken  off.  From  the 
tissue  paper  the  map  can  be  put  on  any  other  paper. 

The  engineers  of  Napoleon's  army  frequently  made  maps  of  the 
localities  they  were  about  to  pass  ;  indeed  it  is  a  military  principle 
never  to  go  into  an  unknown  locality,  except  in  cases  of  absolute 
necessity. 

This  subject  naturally  leads  us  to 

MARINE      SURVEYING. 

Marine  Surveying  is  too  extensive  a  subject  to  be  fully  investi- 
gated in  any  work  like  this.  We  shall  only  explain  how  to  find 
shoals,  rocks,  and  turning  points  hi  a  channel,  by  ranging  to  objects 
on  shore. 

In  trigonometrical  surveying,  on  shore,  the  observer  is  supposed 
to  take  his  angles  from  the  extremities  of  a  base  line,  but  hi  trigono- 
metrical surveying  on  water,  the  observer  can  take  his  angles  only 
from  single  points  which  may  be  connected  together  by  distant  base 
lines  on  the  shore. 

Important  points  along  the  shore  are  determined  by  taking  lati- 
tude and  longitude,  and  intermediate  places,  by  regular  land  sur- 
veying. 

The  localities  of  rocks  and  shoals  are  also  determined  by  astron- 
omical observations,  establishing  latitude  and  longitude,  in  case  no 
land  is  in  sight,  or  they  are  far  from  the  shore  ;  but  in  the  vicinity 
of  the  land,  the  determination  of  a  point  is  commonly  effected  by  the 
three  poi*it  problem. 


MARINE    SURVEYING.  153 

The  three  point  problem  is  the  determination  of  any  point  from 
observations  taken  at  that  point,  on  three  other  distant  points  where 
the  distances  of  these  three  points  from  each  other  are  known. 

It  is  immaterial  how  those  points  are  situated,  provided  the  three 
points  and  the  observer  are  not  in  the  same  right  line,  the  middle 
one  may  be  nearest  or  most  remote  from  the  observer,  or  two  of 
them  may  be  in  one  right  line  with  the  observer,  or  all  three  may  be 
in  one  right  line,  provided  the  observer  be  not  in  that  line.  The 
following  example  will  illustrate  the  principle. 

Coming  from  sea,  at  the  point  D,  I  observed  two  headlands,  A 
and  B,  and  inland  0,  a  steeple,  which  appeared  between  the  head- 
lands. I  found,  from  a  map,  that  the  headlands  were  5.35  miles  from 
each  other  ;  that  the  distance  from  A  to  the  steeple  was  2.8  miles, 
and  from  B  to  the  steeple  3.47  miles  ;  and  I  found  with  a  sextant, 
that  the  angle  ADC  was  12°  15',  and  the  angle  BDQ  15°  30'. 

steeple. 

If  the  direction  of  AB  is  known,  the  direction  of  A  C  is  equally 
well  known. 

The  case  in  which  the  three  objects,  A,  C, 
and  B,  are  in  one  right  line  may  require  illus- 
tration. 

At  the  point  A,  make  the  angle  BAE=.  the 
observed  angle  CDB,  and  at  B,  make  the  angle 
ABE=  the  observed  angle  ADO. 

Describe  a  circle  about  the  triangle  ABE, 
join  E  and  C,  and  produce  that  line  to  the  circumference  in  D, 
which  is  the  point  of  observation.  Join  AD,  BD.  The  angle 
ADB  is  the  sum  of  the  observed  angles,  and  AEB  added  to  it 
must  make  180°. 

The  Trigonometrical  Analysis. —  In  the  triangle  ABE,  we  have 
the  side  AB  and  all  the  angles,  AE  and  EB  can  therefore  be  com- 
puted. 

In  the  triangle  AEG,  we  now  have  A  C,  AE,  and  the  angle  CAE, 
from  which  we  can  compute  A  CE,  then  we  know  A  CD. 

Now  in  the  triangle  A  CD,  we  have  A  C  and  all  the  angles,  whence 
we  can  find  AD  and  CD. 


154  SURVEYING. 

When  the  bearing  of  the  base  line  on  shore  is  known,  as  it  gen- 
erally is,  and  the  bearings  to  its  extremities,  or  even  to  one  extrem- 
ity, are  taken,  the  triangle  is  known  at  once. 

A  pilot  guides  a  vessel  in  and  out  of  a  port  by  ranging  lines  on 
the  shore,  minutely  or  approximately,  as  the  case  may  require. 

We  will  illustrate  this  by  a  figure.  Let  the  deep  shaded  lines 
represent  the  shores,  A  a  light  house  on  a  rocky  promontory.  B 
another  prominent  object  on  the  opposite  shore  ;  the  position  of  the 
arrow  indicates  the  direction  north  and  south. 

The  faint  dotted  lines  represent  the  boundaries  of  shoal  water, 
over  which  it  would  not  be  prudent,  if  even  possible,  to  conduct 
a  vessel.  The  line  a,  b, 
d,  JE,  the  center  of  the 
ship  channel.  All  the  pi- 
lots know  that  a  line  from 
A  to  a,  which  is  nearly 
east  south  east,  runs  safe 
to  the  open  sea,  after 
passing  the  shoal  coast 
near  n. 

Now  suppose  a  pilot 
boards  a  ship  coming  hi  from  sea,  sufficiently  far  from  the  coast,  he 
directs  her  sailing  so  as  to  bring  the  light  house  at  A  to  bear  west 
north-west.  He  then  sails  toward  the  light  house  until  he  finds  the 
object  B  bearing  due  north  of  him.  He  then  knows  that  the  ship 
must  be  near  a,  the  mouth  of  the  channel. 

He  continues  the  same  course,  and  knows  when  he  is  about  half 
way  from  a  to  b  by  the  two  objects,  C  and  D,  appearing  in  the  same 
line.  When  C  and  D  become  fairly  open,  and  C  nearly  north, 
and  B  not  quite  north-east,  he  is  then  at  the  turning  point  b  of  the 
channel.  His  course  is  then  north,  a  little  to  the  west,  until  the 
ship  is  nearly  in  a  line  between  the  two  objects  A  and  B.  From 
thence,  west  north-west  takes  the  ship  directly  through  the  proper 
channel  into  the  harbor. 


ft  .        \ 

LEVELING.  155 


C  HA  P  T  E  R    VIII. 

LEVELING. 

Two  or  more  points  are  said  to  be  on  a  level,  when  they  are 
equally  distant  from  the  center  of  the  earth,  or  when  they  are  equally 
distant  from  a  tranquil  fluid,  situated  immediately  below  them.  A 
level  surface  on  the  earth,  is  nearly  spherical,  and  is  not  a  plane  ; 
it  is  everywhere  perpendicular  to  a  plumb  line. 

Any  small  portion  of  a  true  level  surface,  cannot  be  distinguished 
from  a  plane ;  and,  therefore,  when  observations  are  taken  in  respect 
to  level,  within  short  distances  of  each  other,  the  spherical  form  of 
the  earth  is  disregarded,  and  the  level  treated  as  a  plane.  But  when 
any  considerable  portion  of  surface  is  taken  into  account,  the  curva- 
ture of  the  earth's  surface  must  be  considered. 

The  apparent  level,  at  any  point  on  the  earth,  is  a  tangent  plane, 
touching  the  earth  at  that  point  only,  and  the  true  level  is  below  this, 
and  the  distance  below,  depends  on  the  distance  from  the  tangent 
point. 

Let  T  be  any  point  on  the  surface  of  the 
earth,  at  right  angles  to  the  plumb  line  from 
this  point  is  the  plane  or  apparent  level  ATJS; 
but  the  true  level,  or  the  surface  of  standing 
water,  would  be  the  curved  surface  GTH. 

The  distance  A  0,  depends  on  the  distance 
AT,  and  the  radius  of  the  earth  CG  or  CT. 

From  #,  draw  GD,  at  right  angles  to  A  C ; 
then  the  two  triangles  ATC,  AGD,  are  equi- 
angular and  similar,  and  give  the  proportion 
CT-.TA  \\DO-.  GA. 

Practically.  — TA  is  a  very  short  distance  compared  to  CT,  for 
TA,  the  distance  within  which  we  can  take  observations,  is  never 
more  than  two  or  three  miles,  while  the  distance  CT  is  near  4000 
miles  ;  therefore,  CT  is  nearly  equal  to  CA,  consequently,  DA  is 
nearly  equal  to  DG,  so  near  that  we  shall  call  it  equal. 

Observe  that  TD=DG;  hence 


156  SURVEYING. 

Now,  in  the  preceding  proportion,  in   the  place  of  DG,  put  its 
equal  %TA,  and  we  shall  have, 

CT:TA::     TA  :  GA. 


Whence,         W--,        AfejP-,  (1) 


That  is,  3^0  square  of  the  distance,  divided  by  the  diameter  of  the 
eartht  is  the  distance  between  tJie  apparent  and  the  true  level. 

We  can  arrive  at  the  same  result  by  the  direct  application  of  the 
36th  proposition  of  the  third  book  of  Euclid,  or  by  the  application  of 
theorem  18,  third  book  of  Robinson's  Geometry. 

Because  A  is  a  point  without  a  circle,  and  A  7*  touches  the  circle, 
we  must  have 


But  2  CG,  which  is  the  diameter  of  the  earth,  cannot  be  essentially 
or  appreciably  increased,  by  the  addition  of  AG,  which  is  at  most, 
but  a  few  feet,  therefore,  A  G  within  the  parentheses,  may  be  sup- 
pressed without  making  any  appreciable  error.  Then  divide  by  2  CG. 

AT2 

Whence,  AG=——,  the  same  as  before. 

2C7Cr 

If  we  take  one  mile  for  the  distance  TA,  the  value  of  GA  will 
be  TJr_=8.001  inches. 

By  comparing  equations  (  1  )  we  perceive  that, 
GA  :ga:  :  TA*  :  Ta* 

That  is,  The  corrections  for  apparent  levels,  are  in  proportion  to  the 
squares  of  the  distances. 

The  correction  for  one  mile  is  8.001  inches;  what  is  it  for  10  miles? 
Ans.  It  is  a;  inches  ;  then  we  have  the  following  proportion, 

8.001  :  a:  :  :  I2  :  102  x—  soo.l  inches. 

We  have  seen  above,  that  the  correction  for  one  mile  or  80  chains 
distance,  on  an  apparent  level,  is  8.001  inches,  what  is  the  correction 
for  the  distance  of  20  chains  ? 

Let  a?=the  correction  sought,  and  the  solution  is  thus, 
8.001  :  x  :  :  (80)*  :  (20)2 

:  :     42    :      I2        a?=0.500  inches. 


LEVELING. 


157 


In  this  manner,  the  following  table  was  computed. 

Table  showing  the  differences  in  inches,  between  the  true  and  apparent 
level,  for  distances  between  1  and  100  chains. 


Cb's. 

In's. 

Ch's. 

In'e. 

Ch's. 

In's. 

Ch's. 

In's. 

1 

.001 

"26" 

.845 

51 

3.255 

76 

7.221 

2 

.005 

27 

.911 

52 

3.380 

77 

7.412 

3 

.011 

28 

.981 

53 

3.511 

78 

7.605 

4 

.020 

29 

1.051 

54 

3.645 

79 

7.802 

5 

.031 

30 

.125 

55 

3.785 

80 

8.001 

6 

.045 

31 

.201 

56 

3.925 

81 

8.202 

7 

.061 

32 

.280 

57 

4.061 

82 

8.406 

8 

.080 

33 

.360 

58 

4.205 

83 

8.612 

9 

.101 

34 

.446 

59 

4.351 

84 

8.832 

10 

.125 

35 

.531 

60 

4.500 

85 

9.042 

11 

.151 

36 

.620 

61 

4.654 

86 

9.246 

12 

.180 

37 

.711 

62 

4.805 

87 

9.462 

13 

.211 

38 

.805 

63 

4.968 

88 

9.681 

14 

.245 

39 

.901 

64 

5.120 

89 

9.902 

15 

.281 

40 

2.003 

65 

5.281 

90 

10.126 

16 

.320 

41 

2.101 

66 

5.443 

91 

10.351 

17 

.361 

42 

2.208 

67 

5.612 

92 

10.587 

18 

.405 

43 

2.311 

68 

5.787 

93 

10.812 

19 

.451 

44 

2.420 

69 

5.955 

94 

11.046 

20 

.500 

45 

2.531 

70 

6.125 

95 

11.233 

21 

.552 

46 

2.646 

71 

6.302 

96 

11.521 

22 

.605 

47 

2.761 

72 

6.480 

97 

11.763 

23 

.661 

48 

2.880 

73 

6.662 

98 

12.017 

24 

.720 

49 

3.004 

74 

6.846 

99 

12.246 

25 

.781 

50 

3.125 

75 

7.032 

100 

12.502 

This  table  is  of  little  or  no  practical  use,  for  levelers  rarely  take 
sight  to  a  greater  distance  than  10  chains,  and  at  that  distance  the 
correction  is  only  one-eighth  of  an  inch,  and  if  they  put  the  level 
midway  between  two  stations,  they  annihilate  the  corrections 
altogether. 

Suppose  a  level  to  be  placed  at  T,  midway  between  A  and  B ;  the 
instrument  will  show  them  to  be  on  the  same  level,  as  so  they  really 
are,  for  they  are  at  equal  distances  from  the  center  of  the  earth  ;  but 
if  the  observations  were  taken  in  reference  to  A  and  a,  the  apparent 
level  would  not  show  equal  distances  from  the  center  of  the  earth, 
and  a  correction  must  be  applied,  if  the  difference  of  distances  is 
more  than  four  or  five  chains. 


158  SURVEYING. 

To  comprehend  the  whole  subject,  we  must  now  describe  the 
modern 


SPIRIT     LEVEL. 


The  figure  before  us  represents  this  useful  instrument,  apart  from 
its  tripod. 

Its  principal  parts  are  the  telescope  AJS,  to  which  is  attached  the 
leveling  tube  GD\  the  telescope  rests  in  a  bed,  which  is  supported 
by  posts  yy,  called  the  y's ;  EE  is  a  firm  bar,  supporting  the  y's. 
In  S  is  a  socket,  which  receives  the  central  pivot  of  the  tripod 
(which  is  not  here  represented). 

When  the  instrument  is  put  upon  its  tripod,  the  tube  S  can  be 
clasped  on  the  outside,  and  held  firmly  by  a  clamp  screw,  it  can 
then  be  moved  horizontally,  as  minutely  and  readily  as  desired,  by 
means  of  a  tangent  screw. 

The  tripod  contains  a  pair  of  brass  plates,  to  the  lower  one  the 
legs  of  the  tripod  are  firmly  attached,  the  other  plate  moves  in  all 
directions  on  its  center,  and  is  worked  by  four  screws ;  these  are 
called  the  leveling  screws  ;  these  plates  are  purposely  made  small  as 
a  greater  surety  against  bending :  the  four  leveling  screws  are 
placed  at  the  four  quadrant  points  of  the  circle,  and,  with  the  center, 
form  diameters  at  right  angles. 

The  eye-glass  of  the  telescope  is  at  A,  the  object  glass  at  B.  The 
screw  F  runs  out  the  tube  which  holds  the  object  glass,  to  adjust  it 
to  different  distances. 

The  telescope  is  fastened  into  the  y's,  by  the  loops  r  r,  which  are 
fastened  by  the  pins  p  p.  The  telescope  can  be  reversed  in  the  y's, 
by  taking  out  the  pins  p  p  ;  opening  the  loops  r  r ;  taking  up  the 
tube,  turning  it  round,  and  again  placing  it  in  the  y's  ;  then  A  will 


LEVELING.  1.59 

take  the  position  of  B,  and  B  of  A.  The  necessity  of  this  construe- 
tion  will  appear  when  we  describe  the  adjustment. 

At  n  n  are  two  small  screws  that  are  attached  to  a  ring  inside  of 
the  tube ;  this  ring  holds  a  horizontal  spider  line  ;  the  object  of 
the  screw  is  to  elevate  and  depress  that  spider  line. 

At  q  q  (  only  one  q  can  be  seen  in  the  figure  ),  are  two  screws 
that  work  another  ring,  which  holds  a  perpendicular  spider  line, 
which  can  be  moved  to  the  right  and  left  by  the  screws  q  q.  The 
two  spider  lines  show  a  perpendicular  and  horizontal  cross  at  the 
focus  of  the  telescope. 

Before  using  the  instrument  it  must  be  adjusted.  The  adjust- 
ment consists : 

1st.  In  making  the  center  of  the  eye-glass  and  the  intersection  of  the 
spider's  lines  coincide  with  the  axis  of  the  telescope,  and  this  line  is 
called  the  line  of  collimation. 

2nd.  In  making  the  axis  of  the  attached  level,  CD,  parallel  to  the 
line  of  collimation,  in  respect  to  elevation. 

3rd.  In  making  the  attached  level  lie  exactly  in  the  same  direction  as 
the  line  of  collimation. 

To  make  the  first  adjustment,  the  telescope  is  made  to  revolve 
in  the  y's. 

To  make  the  second  adjustment,  there  is  a  screw  a,  which  serves 
to  elevate  and  depress  the  end  of  the  leveling  tube  at  0. 

To  make  the  third  adjustment,  there  is  a  side  screw  5,  which 
drives  the  end  of  the  tube  D  to  the  right  and  left,  as  the  case  may 
require. 

First  Adjustment. —  Plant  the  tripod,  place  the  instrument  upon 
it,  and  direct  the  telescope  to  some  well  denned  and  distant  object. 

Draw  out  the  eye-glass  at  A,  until  the  spider's  lines  are  distinctly 
seen,  then  run  out  the  object  glass  by  the  screw  V  to  its  proper  fo- 
cus, when  the  object  and  the  spider's  lines  will  be  distinct.  Now 
note  the  precise  point  covered  by  the  horizontal  spider's  line. 
Having  done  this,  revolve  the  telescope  in  the  y's  half  round,  when 
the  attached  level  will  be  on  the  upper  side.  See  if  in  this  position 
the  horizontal  spider's  line  appears  above  or  below  the  same  object. 

If  this  line  should  appear  exactly  in  the  same  point  of  the  object 


160  SURVEYING. 

as  before,  this  spider's  line  is  already  in  adjustment,  but  if  it  ap- 
pears above  or  below,  bring  it  half  way  to  the  same  point  by  means 
of  the  screws  n  n,  loosening  the  one  and  tightening  the  other. 

Carry  back  the  telescope  to  its  original  position,  and  repeat 
the  observation,  and  continue  to  repeat  it  until  the  telescope  will 
revolve  half  round  without  causing  the  horizontal  line  to  rise  or 
fall.  This  will  show  that  the  horizontal  line  is  a  diameter  of  the  cir- 
cle of  revolution,  and  not  a  chord  of  it.  Make  the  same  adjust mtnt 
in  respect  to  the  vertical  hair,  and  the  line  of  collimation  is  then 
adjusted. 

Second  Adjustment. —  That  is,  to  make  the  tube  CD  horizontally 
parallel  to  the  line  of  collimation.  Place  the  instrument  properly  on 
its  tripod,  and  bring  the  horizontal  bar  EE  directly  over  two  of  the 
leveling  screws  ;  turn  these  screws  until  the  bubble  d  rests  hi  the 
center  of  the  tube. 

Now  CD  is  on  a  level,  but  we  are  not  able  to  say  that  the  line  of 
sight  through  the  telescope,  that  is  the  line  of  collimation,  is  on  a 
level  also.  To  test  this,  take  out  the  pins  p  p,  open  the  loops  r  r, 
and  take  out  the  telescope  with  its  attached  level,  and  turn  it  end 
for  end,  put  it  back  in  its  bed,  and  put  the  loops  over  and  pin  them 
down.  If  the  bubble  now  rests  in  the  middle,  no  adjustment  is  re- 
quired; if  not,  the  bubble  will  run  to  the  elevated  end.  In  that  case 
the  bubble  must  be  brought  half  way  back  by  the  leveling  screws, 
and  the  other  half  by  the  screw  a. 

Repeat  the  operation  until  the  bubble  will  settle  in  the  middle  of 
the  tube  after  reversing  the  telescope. 

Third  Adjustment. —  The  second  adjustment  being  completed,  re- 
volve the  telescope  in  the  y's,  and  if  the  bubble  continues  in  the 
middle,  the  axis  of  the  telescope  and  the  axis  of  the  tube  CD  lie  in 
the  same  direction,  or  in  the  same  vertical  plane  ;  and  if  they  be  not 
in  the  same  vertical  plane  the  bubble  will  run  to  one  end  or  the 
other  ;  in  that  case  the  side  screw  b  will  remedy  the  defect. 

The  three  adjustments  are  now  made  approximately,  no  one  of 
them  can  be  made  perfectly  while  the  instrument  is  greatly  out  of 
adjustment  in  relation  to  the  others  ;  therefore  commence  anew. — 
Bring  the  bar  EE  over  two  of  the  leveling  screws,  and  level  the 
instrument;  then  turn  it  over  the  two  other  screws  and  level  it  in  that 
direction  also.  Now,  if  we  can  turn  the  instrument  quite  round 


LEVELING.  161 

without  removing  the  bubble  from  the  center  it  is  in  pretty  good  ad- 
justment, but  if  otherwise,  as  is  to  be  expected,  make  all  these 
adjustments  over  again;  they  can  now  be  made  with  much  less 
difficulty. 

It  is  important  that  a  level  should  be  in  as  perfect  adjustment  as 
possible,  but  perfection  in  all  respects  is  almost,  yea,  quite  impossi- 
ble. Yet,  with  a  level  considerably  out  of  adjustment,  we  can  ob- 
tain the  relative  elevation  of  any  two  points,  provided  we  can  set  the 
level  midway  between  them. 

To  illustrate  this,  suppose  the  instrument  placed  at  D,  midway 
between  two  perpendicular  rods  Aa  Bb. 


Let  ab  represent  the  true  horizontal  line,  but  suppose  that  the  in- 
strument is  so  imperfect,  or  out  of  adjustment,  that  when  the  level- 
ing tube  CD  is  horizontal,  the  telescope  would  point  out  the  rising 
line  DA,  and  the  rise  would  be  Aa.  On  turning  the  instrument 
round  and  sighting  to  B,  the  rise  must  be  the  same  as  in  the  oppo- 
site direction :  for  the  distance  is  the  same,  therefore  A  and  B  are  as 
truly  on  a  level  with  each  other  as  a  and  b. 

By  this  problem,  practical  men  complete  the  second  adjustment  of 
the  instrument.  They  make  the  three  adjustments  as  just  explained, 
as  accurately  as  possible.  They  then  measure,  very  carefully 
the  distance  between  two  stations,  as  E  and  F,  and  set  the  in- 
strument exactly  midway  between  them  as  represented  in  the  last 
figure. 

They  then  level  the  instrument  (  that  is  the  tube  CD ),  and 
find  the  difference  of  the  levels  between  E  and  F  (  two  pegs  driven 
into  the  ground  ). 

Now,  suppose  AE  measures  on  the  rod,  -         4.752  feet. 

And  BF        '<          "        "  -     6.327  feet. 

Then  E  is  above  F 1.575  feet. 

11 


162  SURVEYING. 

They  now  bring  the  level  near  to  one  of  the  stations  as  E,  and 
level  it  very  accurately,  and  sight  to  the  rod  AE. 

Now,  suppose  the  target  stands  at     -       -         -         5.137  feet. 

To  this  add  -        -         -     1.575  feet. 

6.712 

The  rod  man  now  goes  to  the  station  F,  puts  his  target  on  the  rod 
exactly  at  6.712,  and  the  telescope  is  turned  upon  it,  and  the  hori- 
zontal spider's  line  ought  to  just  coincide  with  the  target,  and  will,  if 
the  instrument  is  in  perfect  adjustment ;  if  it  is  not,  the  error  is  taken 
out  by  the  screws  n  n.  If  the  error  was  but  slight,  as  in  such  cases 
it  always  is  with  good  instruments,  the  adjustment  is  as  complete  as 
it  can  be  made. 

With  the  level  there  must  be 

A    BOD. 

The  rod  is  commonly  ten  feet  long,  and  divided  into  tenths  and 
hundredths,  some  have  also  a  vernier  scale  which  hi  effect  subdivides 
to  thousandths.  The  target  slides  up  and  down  the  rod,  and  car- 
ries the  vernier  on  the  back  of  the  rod ;  the  target  has  equal  alternate 
portions  painted  black  and  white  for  contrast. 

A  party  to  take  the  necessary  levels  on  the  line  of  a  railroad  or 
canal,  after  the  stations  are  measured  off,  should  consist  of  a  leveler, 
and  assistant  leveler,  a  rod  man,  and  an  axe  man. 

The  leveler  and  assistant  leveler  both  keep  book,  and  sometimes 
the  rod  man  also.  If  there  is  no  assistant  leveler,  the  rod  man  will 
have  an  abundance  of  time  to  keep  book,  and  under  such  circum- 
cumstances  always  does  so.  Under  all  circumstances,  two  persons 
keep  book,  to  have  a  check  on  each  other  and  guard  against  mistakes. 

In  the  field  the  aim  is  to  put  the  level  midway  between  the  two 
stations,  but  they  are  not  particular  about  it  if  the  instrument  is  in 
good  adjustment ;  they  rather  take  the  most  advantageous  spot  to 
sight  from. 

When  the  ground  admits  of  it,  that  is,  sufficiently  level,  two  or 
three  intermediate  stations  are  observed,  as  well  as  the  extreme  back 
and  fore  stations.  The  extreme  back  and  fore  stations  are  called 
changing  stations  ;  at  these  stations  pegs  are  driven  into  the  ground 
by  the  axe  man  for  the  rod  man  to  plant  his  rod,  so  as  to  secure 


LEVELING. 


163 


the  same  point  for  both  the  fore  and  back  sight.  At  the  intermedi- 
ate stations  they  have  no  pegs,  and  are  not  particular  in  any  respect, 
for  all  errors  cancel  each  other. 

The  common  railroad  chain  is  100  feet,  divided  into  100  links ; 
each  link  is  therefore  one  foot.  Levels  are  commonly  taken  at  inter- 
vals of  200  feet,  oftener  if  the  ground  is  very  uneven,  but  a  station 
is  considered  200  feet,  and  the  number  of  the  station  multiplied  by 
200  gives  the  number  of  feet  from  the  commencement. 

The  field  book  is  kept  thus  : 

R  S.  means  back  sight,  F.  S.  fore  sight,  A.  ascent,  D.  descent,  T.  total 
elevation  above  a  common  base. 

N.  B. —  When  the  back  sight  is  less  than  the  fore  sight,  the 
ground  is  descending,  and  the  converse. 


Sta. 

B.  S. 

F.  S. 

A. 

D. 

Total. 

100 

0 

4.32 

7.21 

2.89 

97.11 

1 

5.52 

8.17 

2.65 

94.46 

2 

*9.18 

6.27 

2.91 

97.37 

3 

6.27 

6.12 

0.15 

97.52 

4 

6.12 

3.76* 

2.36 

99.88 

5 

9.81 

11.62 

1.81 

98.07 

6 

8.47 

9.02 

0.55 

97.52 

7 

2.64 

8.91 

6.27 

91.25 

8 

1.07 

7.38 

6.31 

84.94 

9 

4.29 

5.32 

1.03 

83.91 

10 

5.32 

4.85 

0.47 

84.38 

11 

4.85 

3.17 

1.68 

86.06 

12 

8.22 

1.53 

7.31 

93.37 

Thus  we  go  through  the  whole  line.  We  commenced  with  the 
constant  100,  but  this  is  arbitrary;  the  object  of  taking  a  constant  is 
to  avoid  the  minus  sign,  that  is,  getting  below  our  ruled  paper  in 
making  a  profile  of  the  vertical  section. 

Where  the  line  is  to  be  generally  ascending,  we  assume  a  small 
constant,  where  generally  descending  a  large  one  ;  taking  care  in  all 
cases  to  have  it  so  large  as  not  to  run  it  out. 

At  each  and  every  section  we  know  exactly  how  much  we  are 
above  or  below  the  constant  base,  and  the  exact  ascent  or  descent 
from  any  one  station  to  any  other. 

The  following  diagram  represents  a  vertical  section  of  the  ground 


164  SURVEYING. 

where  these  levels  were  taken,  with  the  exception  of  the  exaggera- 
tion of  the  roughness  caused  by  the  difference  of  scales  for  the  base 
and  perpendicular  lines.  From  one  station  to  another  is  200  feet  ; 
we  have  made  10  feet  occupy  more  space  in  the  perpendicular  direc- 
tion than  400  feet  does  in  a  horizontal  direction.  We  do  this  to 
show  more  clearly  where  any  particular  grade  will  enter  the 
ground,  and  how  much  it  is  necessary  to  cut  or  fill  at  any  particular 
point. 

The  zigzag  line  from  100  of 
altitude  to  86,  represents  the 
surface  of  ground,  and  suppose 
that  we  wished  to  reduce  it  to 
a  regular  grade  so  as  to  remove 
as  little  earth  as  possible.  By 
the  mere  exercise  of  judgment,  U 
we  conclude  to  run  the  grade  between  0  station  and  11,  from  98  to 
92  (  but  the  propriety  of  this  conclusion  would  depend  on  the  con- 
tour of  the  ground  on  each  side  of  these  stations  ).  The  direct  line 
a  b  drawn,  shows  that  the  grade  runs  out  of  the  ground  at  station 

I,  we  must  fill  in  about  2^  feet  at  station  2,  the  grade  runs  into  the 
ground  again  at  about  80  feet  before  we  come  to  station  3.     At  sec- 
tion 4  the  cutting  must  be  a  little  over  2  feet,  at  station  5  a  little 
over  5  feet,  at  7,  2  feet,  and  runs  out  of  the  ground  midway  be- 
tween stations  7  and  8. 

At  stations  9  and  10  we  must  fill  in  about  8  feet,  and  so  on ;  the 
depth  of  cutting  or  filling  is  obvious  at  every  station. 

If  the  contour  of  the  ground  beyond  1 1  was  generally  level  or 
descending,  we  would  change  the  grade  at  station  7  and  render  it 
more  descending,  so  as  to  make  less  filling  up  at  stations  9,  10,  and 

II.  In  the  adoption  of  grades  for  a  railroad,  an  engineer  has  great 
scope  to  exercise  his  judgment. 

In  the  representation  here  made,  ab  appears  like  a  steep  grade, 
but  it  is  not ;  it  could  scarcely  appear  on  tlu?  ground  other  than  a 
level,  for  the  difference  is  only  6  feet  in  2200  feet  of  distance,  which 
is  at  the  rate  of  14T\  feet  to  the  mile. 

Engineers  can  freely  vary  their  grade,  while  it  keeps  under  18  or 
20  feet  to  the  mile  ;  but  they  submit  to  a  great  deal  of  cutting  and 
filling,  before  they  establish  a  grade  over  40  feet  to  the  mile. 


LEVELING.  165 

CONTOUR     OF     GROUND, 

Contour  of  ground  is  shown  on  maps,  by  marking  where  parallel 
planes  run  out  on  the  surface. 

We  shall  give  only  the  general  principle. 

Let  A  be  the  top  of  a  hill,  whose  contour  we  wish  to  delineate; 
measure  any  convenient  line  as  AB,  up  or  down  the  hill,  and  by 
the  level  or  theodolite,  ascertain  the  relative  elevations  of  a,  b,  c,  d, 
&c.,  as  many  planes  as  we  wish  to  represent. 

At  a,  place  the  level  or  theodolite,  and  level  it  ready  for  observa- 
tion ;  measure  the  height  of  the  instrument,  and  put  the  target  on 
the  rod  at  that  height. 

Send  the  rod -man  and 
axe -man  round  the  hill,  on 
the  same  level  as  the  in- 
strument. Let  the  rod-man 
set  the  rod ;  the  leveler 
will  sight  to  it  through  the 
telescope,  and  if  the  tar- 
get is  below  the  level,  he 
will  motion  the  rod-man  up  the  hill,  if  too  high,  down  the  hill ;  at 
length  he  will  get  the  same  level,  and  there  the  axe-man  will  drive 
a  stake.  In  the  same  manner  we  will  establish  another  stake  further 
on ;  and  thus  proceed  from  point  to  point.  To  get  round  the  hill, 
it  may  be  necessary  to  move  the  instrument  several  tunes.  The 
plane  thus  established,  is  represented  by  the  curve  am. 

In  the  same  manner,  by  placing  the  instrument  at  b,  we  can 
establish  the  next  plane  bn. 

Then  the  next,  and  the  next,  as  many  as  we  please.  Where  the 
hill  is  more  steep,  two  of  these  parallel  planes  will  be  nearer  together 
in  the  figure;  where  less  steep,  they  will  appear  at  a  greater  distance 
asunder,  and  this,  with  the  proper  shading,  will  give  a  true  repre- 
sentation of  the  ground. 


166  SURVEYING. 

ELEVATIONS     DETERMINED     BY     ATMOSPHERIC     PRES- 
SURE,     ASINDICATED      BY      T  H  E    B  A  R  O  M  E  T  E  R  . 

The  higher  we  ascend  above  the  level  of  the  sea,  the  less  is  the 
atmospheric  pressure  (other  circumstances  being  the  same),  and 
therefore  we  can  determine  the  ascent,  provided  we  can  accurately 
measure  the  pressure,  and  know  the  law  of  its  decrease. 

As  this  work  is  designed  to  be  educational  as  well  as  practical, 
wo  shall  here  make  an  effort  to  explain  the  philosophy  of  the  pro- 
blem, in  such  a  manner,  as  to  force  it  upon  the  comprehension  of  the 
learner. 

The  pressure  of  the  atmosphere  at  any  place,  is  measured  by  the 
height  of  a  column  of  mercury  it  sustains  in  the  barometer  tube. 

It  is  found  by  experiment,  that  air  is  compressible, 
and  the  amount  of  compression  is  always  in  pro- 
portion to  the  amount  of  the  compressing  force. 

Now,  suppose  the  atmosphere  to  be  divided  into 
an  indefinite  number  of  strata,  of  the  same  thick- 
ness, and  so  small  that  the  density  of  each  stratum 
may  be  considered  as  uniform. 

Commence  at  an  indefinite  distance  above  the 
surface  of  the  earth,  as  at  A,  and  let  w  represent 
the  weight  of  the  whole  column  of  atmosphere 
resting  on  A.  Let  the  small  and  indefinite  dis- 
tances between  AB,  BC,  CD,  <fec.,  be  equal  to  each  other,  and  we 
shall  call  them  units  of  some  unknown  magnitude. 

The  weight  of  the  column  of  atmosphere  supposed  to  rest  on  B, 
is  greater  than  w,  by  some  indefinite  part  of  w,  say  the  wth  part. 

Then  the  weight  on  B,  must  be  expressed  byf  w~}--  jor  (  M'     \w. 

In  the  same  manner,  the  weight  or  pressure  resting  on  C,  must  be 
the  weight  above  B,  increased  by  its  wth  part  ;  that  is,  it  must  be 


•  which  by  addilion  is 

In  the  same  manner,  we  find  that  the  weight  resting  on  D,  must 
^    '     '  w,  and  so  on.  For  the  sake  of  perspicuity,  we  recapitulate. 


LEVELING.  167 

The  pressure  on  A  is        w.      Units  from  A  0 


"        on  B  is  -w    "          "      1 

\   n    J 

«        on  0  is  (±tl)!  w   ••          "2 
»* 

«        on  D  is  fo+1)3  w    "          "3 


"        on  E  is  (*"l"*)4  w    " 


"4 


on        s 

«* 

<fcc.  &c.  &o.  <fec. 

Here  we  observe  the  series  which  represents  the  pressure  of 
atmosphere,  at  the  different  points  A,B>  (7,&c.,  is  a  series  in  geometrical 
progression,  and  it  corresponds  with  another  series  in  arithmetical 
progression;  therefore,  by  the  nature  of  logarithms,  the  numbers  in 
the  arithmetical  series,  may  be  taken  as  the  logarithms  of  the  num- 
bers in  the  geometrical  series. 

But  this  system  of  logarithms,  may  not  be  hyperbolic  nor  tabular, 
indeed  it  is  neither  ;  the  base  of  this  system  is  as  yet  unknown,  but 
our  investigations  will  soon  lead  to  its  discovery. 

Now,  let  the  number  of  units  from  A  to  S  (the  surface  of  the 
sea),  or  to  the  lower  of  two  stations,  be  represented  by  x,  then  the 

expression  for  the  pressure  of  the  air  would  be(  -     -  Yt0,  but  this 

is  neither  more  nor  less  than  the  weight  of  the  column  of  mercury 
in  the  barometer,  which  is  sustained  by  this  pressure. 

By  calling  this  b,  and  designating  the  logarithms  of  this  unknown 
system  by  L',  we  shall  have 

L'b=x  (1) 

Taking  y  to  represent  the  number  of  units  from  A  to  F,  and  bt  to 
represent  the  pressure  of  the  air  at  that  point,  we  shall  have 

L'b,=y  (2) 

Subtracting  (2)  from  (1),  we  shall  have 


This  is,  a  certain  peculiar  logarithm  of  the  barometer  column  at  the 
lower  station,  diminished  by  the  logarithm  of  the  barometer  at  the 


168  SURVEYING. 

upper  station  will  give  the  difference  of  levels  between  the  two  sta- 
tions. But  still  all  is  indefinite  and  unknown,  because  we  know 
nothing  of  these  logarithms. 

In  algebra,  we  learn  that  the  logarithms  of  one  system  can  be 
converted  into  another  by  multiplying  them  by  a  constant  multiplier 
called  the  modulus  of  the  system,  therefore, 

Assume  Z  to  be  the  modulus  or  constant  that  will  convert  com- 
mon logarithms  into  these  peculiar  logarithms. 

Then,  Z(log.  £—  log.  b,)=SV  (3) 

Here,  log.  b  denotes  the  common  logarithms  of  the  barometer 
column. 

Equation  (3)  is  general,  and  determines  nothing  until  we  know 
SV'm  some  particular  case. 

Taking  S  V  some  known  elevation,  and  observing  the  altitude  of 
the  barometer  column  at  both  stations,  and  then  equation  (3)  will 
give  Z  once  for  all. 

Putting  h  to  represent  the  known  elevation,  and  we  have,  in  general, 

Z=log.  £—  log.  bt  (4) 

Example.  —  At  the  bottom  and  top  of  a  tower,  whose  height  was 
200  feet,  the  mercury  stood  in  the  barometer  as  follows. 

At  the  bottom,  -         -  -  29.96  inches  =5 

At  the  top,  -  -      29.74  inches  =£, 

the  temperature  of  the  air  being  49°  of  Fahrenheit's  thermometer. 


log.  29.96-  log.  29.  7401 
"  But  this  multiplier  is  constant  only  when  the  mean  temperature 
of  the  air  at  the  two  stations  is  the  same  ;  and  for  a  lower  tempera- 
ture the  multiplier  is  less,  and  for  a  higher  it  is  greater.  A  cor- 
rection, however,  may  be  applied  for  any  deviation  from  an  assumed 
temperature,  by  increasing  or  diminishing  (  according  as  t/ie  tempera- 
ture is  higher  or  lower)  the  approximate  height  by  its  449th  part  for  each 
degree  of  Fahrenheit's  thermometer.  We  can  moreover  change  the 
multiplier  to  a  more  convenient  one  by  assuming  such  a  tempera- 
ture as  shall  reduce  this  number  to  60000  instead  of  62500.  Now 
62500  exceeds  60000  by  its  25th  part;  and,  since  1°  causes  a 
change  of  one  449th  part,  the  proportion 
*b  :  1°  :  :  ^  :  17.9, 


LEVELING.  169 

gives  18°  nearly  for  the  reduction  to  be  made  in  the  temperature  of 
the  air  at  the  time  of  the  above  observations,  in  order  to  change  the 
constant  multiplier  from  62500  to  60000,  or  to  10000,  by  calling 
the  height  fathoms  instead  of  feet.  Thus,  instead  of  the  thermometer 
standing  at  49°,  we  may  suppose  it  to  stand  at  49° — 1 8°  or  31°;  and 
then, we  take  10000  as  the  multiplier,  and  apply  a  correction  addi- 
tive for  the  18°  excess  of  temperature." 

The  same  observations,  for  example,  being  given,  to  find  the 
height  of  the  tower. 

29.96          -         -        log.         -  1.47654 

29.74     -        -         -     log.     -         •  1.47334 

Diff.  of  log.  -         -         0.00320 

Multiplier         -        -        -      10000 
Product         -  32 

Then  the  height  of  the  tower  is  32  fathoms,  or  32X6=192  feet, 
on  the  supposition  that  the  temperature  of  the  air  was  31°  in  place 
of  49°.  But  it  being  49°,  we  must  increase  192  by  its  T^  part  for 
each  degree  above  31°,  that  is,  by  J-fa  or  ^  nearly  of  its  approxi- 
mate height,  which  gives  nearly  8  feet  to  add  to  192,  making  200 
feet  for  the  height  of  the  tower. 

The  same  method  is  applicable  to  other  cases  whatever  be  the 
temperature  of  the  air  at  the  two  stations,  provided  it  be  the  same 
or  nearly  the  same  at  both  stations,  or  provided  we  take  the  mean 
temperature  of  the  two  stations.  We  can  find  the  difference  of 
levels  of  two  stations  to  considerable  accuracy  by  the  following 

RULE.  —  1st.  Take  the  difference  of  the  logarithms  of  the  two  baro- 
meter columns,  and  remove  the  decimal  point  four  places  to  the  right. 
This  is  the  approximate  difference  of  levels  in  fathoms. 

2d.  Add  T|¥  of  the  approximate  height  for  each  degree  of  tempera- 
ture above  31°,  and  SUBTRACT  the  same  for  each  degree  below  31°  ;  the 
result  cannot  be  far  from  the  truth. 

EXAMPLES. 

1.  The  barometer  at  the  base  of  a  mountain  stood  at  29.47  inches, 
and  taking  it  to  the  top,  it  fell  to  28.93  inches. 

The  mean  temperature  of  the  air  was  51°.  What  was  the  height 
of  the  mountain  in  feet  ?  Ans.  503.34  feet. 


170  SURVEYING. 

29.47  log.    -  1.469380 

28.93  log.  -     1.461348 

O.Q08032 

Approximate  height  in  fathoms,  80.32. 
Correction. —  Add  £fr  of  80.32  to  itself,  that  is,  add  3.57. 
Height,  in  fathoms,    -  83.89 

Multiply  by  6 

Height  in  feet,  603.54 

The  average  height  of  the  barometer,  at  the  level  of  the  sea,  is 
30.09  inches ;  and  now  if  we  know  the  average  height  of  the  baro- 
meter at  any  other  place,  and  the  average  temperature,  it  is  equiva- 
lent to  knowing  the  elevation  of  the  latter  place  above  the  level  of 
the  sea. 

For  example,  the  mean  height  of  the  barometer  at  Albany  Academy 
is  29.96,  and  the  mean  temperature  is  49°.  How  high  is  the 
academy  above  tide  water  ? 


Ans.  117.3  feet. 

30.09  log. 
29.96  log. 

1.478422 
1.476542 

49° 
31 

0.001880 

18° 

Approximate  height  18.80  fathoms. 

Add  Jfy  or  A-  75 

19.55X6=117.3  feet. 

2.  The  average  height  of  the  barometer  at  Amenia    Seminary 
in  Duchess,  Co.,  New  York,  is  stated  in  the  Regents'  report  at  29.81 
inches  :  average  temperature  49°.     What  is  the  height  of  that  point 
above  tide  water  ? 

Ans.  253.32  feet. 

3.  The  mean  height  of  the  barometer  at  Pompey  Academy,  On- 
ondagua  Co.,  New  York,  is  28.13,  corrected  and  reduced  to  32° 
Fahr.     What  then  is  the  elevation  of  that  locality  ? 

Ans.  1755  feet. 
Others  have  made  it  1745  feet. 

4.  From  various  observations  on  the  summit  of  Mount  Washing- 
ton, in  New  Hampshire,  the  mean  height  of  the  barometer  there  is 
24.20,  mean  temperature  at  the  times  of  observation  was  about  65° 
Fahr. 


LEVELING.  171 

Now  admitting  that  the  mean  temperature  at  the  surface  of  the 
sea,  in  the  same  latitude  must  have  been  75°,  which  would  make 
the  mean  temperature  between  the  two  stations  70°,  what  then  is 
the  height  of  Mount  Washington  above  the  sea  ? 

Ans.  6170  feet,  nearly. 

By  some  observaiions  the  elevation  is  estimated  at  6496  feet,  by 
others  at  6290  feet. 

5.  Lieutenant*  Fremont,  in  his  narrative  of  the  exploring  expedi- 
tion across  the  Rocky  Mountains,  page  45,  under  date  of  July  13th, 

1842,  states  his   latitude  at  41°  8' 31",  longitude  104°  39'  37", 
height  of  the  barometer  24,86  inches,  attached  thermometer  68° ; 
what  was  his  elevation  above  the  sea  ?  Ans,  5389.2  feet. 

REMARK. — The  author  states  his  elevation  at  5449.  feet.  As  he  does  not  state 
the  temperature  of  the  air  by  the  detached  thermometer,  we  know  not  what 
correction  he  made.  These  solitary  barometrical  observations  are  more  or  less 
valuable,  according  to  the  settled  or  unsettled  state  of  the  weather.  A  person 
of  experience  and  good  judgment  in  such  matters,  Jike  Fremont,  would  not  of 
course  record  the  inapplicable  observations. 

6.  Lieut.  Fremont,  in  his  journal,  page  104,  under  date  of  August 
15th,  1842,  when,  as  he  supposed,  on  the  highest  point  of  the  Rocky 
Mountains,  observed  the  barometer  to  stand  at  18.29  inches,  and  the 
attached  thermometer  at  44°  f ;  what  was  the  elevation  above  the 
level  of  the  sea  ?  Ans.   13522  feet. 

Fremont  estimates  the  elevation  at  13570  feet. 

This  shows  that  he  estimated  the  mean  temperature  above  60°, 
and  no  doubt  a  similar  cause  made  the  difference  in  the  result  of 
the  previous  example. 

7.  On  page  140,  of  Fremont's  Journal,  under  date  of  July  12th, 

1843,  he  says  ;  "  The  evening  was  tolerably  clear,  with  a  tempera- 
ture at  sunset,  of  63°.     Elevation  of  the  camp,  7300  feet." 

Taking  the  mean  temperature  of  the  two  stations,  the  sea  and  his 

*  Lieutenant  was  his  proper  title  at  this  time. 

t  If  the  sea  were  at  the  base  of  the  mountain,  the  temperature  at  the  lower 
station  would  no  doubt  be  as  high  as  60°.  Making  this  supposition,  the  mean 
temperature  of  the  two  stations  would  be  50°.  We  therefore  take  50°  as  the 
mean  temperature. 


172  SURVEYING. 

place  of  observation,  at  67°,  what  must  have  been  the  height  of  his 
barometer  ?  Ans.  23.21  inches. 

Kepresent  the  approximate  elevation  by  y>  then 

Or,  y=6738.14 


449 

Divide  y  by  6,  which  gives   1126.35.     Divide  this  by  10000. 
Then,  let  x  represent  the  altitude  of  the  barometer  column. 
Whence,         1.478422—  log.  #=0.  11  2635 
Therefore,  log.  x=  1  .365787 

In  the  preceding  examples  we  could  only  be  general  and  approxi- 
mate, we  had  only  the  observations  at  one  station  referred  to  general 
observations  at  the  other  ;  but  our  results  cannot  be  far  from  the 
truth. 

When  the  difference  of  temperatures  at  the  two  stations  is  con- 
siderable, the  result  must  be  affected  by  it. 

When  the  upper  station  is  the  coldest,  which  is  generally  the  case, 
the  mercurial  column  will  be  shorter  than  it  otherwise  would  be,  and 
consequently  indicate  too  great  a  height. 

If  the  temperature  of  the  upper  station  is  taken  for  the  tempera- 
ture of  the  lower,  the  mercurial  column  at  the  lower  station  would 
not  be  high  enough,  and  the  deduced  result  would  be  too  small,  as 
is  the  case  in  example  5. 

The  contraction  of  mercury  being  about  one  10000th  part  for 
each  degree  of  cold,  or,  0.0025  in  a  column  of  25  in.,  it  would 
require  4°  difference  of  temperature,  to  produce  an  effect  amounting 
to  one  division  on  the  scale  of  a  common  barometer,  where  the 
graduation  is  to  hundredths  of  an  inch. 

This  correction  is  combined  with  the  former  in  the  following 
equation,  in  which  1  1'  represents  the  temperature  of  the  air  at  the 
two  stations  ;  t  at  the  lower  station,  q  and  q'  the  temperature  of  the 
mercury,  as  indicated  by  the  attached  thermometer. 

The  fraction  0.00223,  is  equal  to  TJT  nearly  ;  £=the  height 
sought,  b  and  ft,  represent  the  observed  height  of  the  mercurial 
column. 


A-.0000     ,+0.00^3-3!        ,og. 


LEVELING.  173 

Beside  the  corrections  previously  considered,  regard  is  sometimes 
had  to  the  effect  of  the  variation  of  gravity  in  different  latitudes, 
and  at  different  elevations  above  the  earth's  surface.  The  latter, 
however,  is  too  small  to  require  any  notice  in  an  elementary  work. 
The  former  may  be  found  by  mulciplying  the  approximate  height 
by  0.0028371  Xcos.  2  lat.  It  is  additive,  when  the  latitude  is  -less 
than  45°,  and  subtractive  when  greater.  Or  it  may  be  taken  from 
the  following  table. 

Latitude.  Correction. 

0°    -  -   -f-    sT¥  °f  to®  app.  height. 

6°  +     ¥*T 

10°  -        -       -r    -f  rfj 

UP  r       -  .      *  J*T 

20°  -         -         -  ,+  H» 

25°  4-  jh 

30°  -                  -4-  jh 

*'  «tO  i         I 

00  T  TffSfl 

40°     -  -     -fWa* 

45°         -         -  0 

fJQO        _  1 

55°  '  -  T°oV'o 

60°    -  -    —   j±j 

65°  —    si* 

70°    -         -         -    —   ^ 

75°  "   T*T 

80°    -  -    -   rfT 

85°  —    3}T 

90°    -  '    —    TOT 

Given,  the  pressure  of  the  atmosphere  at  the  bottom  of  a  moun- 
tain, equal  to  29.68  in.  of  mercury,  and  that  at  its  summit,  equal  to 
25.28  in.,  the  mean  temperature  being  50°,  to  find  the  elevation. 

Ans.  727.2  fathoms,  or  4363.2  feet. 

The  following  observations  being  taken  at  the  foot  and  summit  of 
a  mountain,  namely, 

at  the  foot,      bar.  29.862  attach,  therm.  78°  detach,  therm.    71° 
at  the  summit,  "     26.137  "  63°  "  65° 

to  find  the  elevation. 

Ans.  612.9  fathoms,  or  3677.4  feet. 


174  SURVEYING. 

It  is  required  to  find  the  height  of  a  mountain  in  latitude  30°,  the 
observations  with  the  barometer  and  thermometer  being  as  follows  ; 
namely, 

at  the  foot,      bar.  29.40  attach,  therm.  50°  detach,  therm.*  43° 
at  the  summit,  "    25.19          "  46°  "  39° 

Ans.  683.27  fathoms,  or  4099.62  feet. 

If  we  assume  any  temperature,  for  instance  45°,  and  the  height 
of  the  barometer  at  the  level  of  the  sea,  at  30.09  niches  ;  we  can 
compute  the  elevation  of  the  point,  where  it  would  be  29.99,  29.89, 
29.79,  29.69,  &c.,  inches ;  and  thus  we  might  form  a  table,  showing 
the  elevations  that  would  correspond  to  any  assumed  height  of  the 
barometer  at  that  temperature.  It  will  be  found,  that  the  first  fall 
of  TV  of  an  inch  will  correspond  to  about  88  feet  in  elevation,  but 
every  subsequent  tenth,  will  require  a  greater  and  greater  elevation. 

*  The  attached  thermometer  measures  the  temperature  of  the  mercury  in  the 
barometer,  and  the  detached  thermometer,  that  of  the  surrounding  air. 


NAVIGATION. 


CHAPTER    I. 

INTRODUCTION. 

NAVIGATION  is  the  art  of  conducting  a  ship  from  one  place  to 
another. 

In  most  works  this  art  is  mixed  up  with  seamanship  and  elementary 
science.  In  this  work,  navigation  will  stand  by  itself — alone  ;  and 
we  shall  presume  that  the  reader  is  properly  prepared  in  elementary 
science. 

This  being  the  case,  it  will  not  be  necessary  to  take  up  time  and 
space  in  giving  definitions  of  latitude,  longitude,  meridian,  horizon, 
&c.,  <fcc.,  the  previous  indispensible  knowledge  of  geography  neces- 
sarily gives  a  knowledge  of  all  these  terms. 

Navigation,  rightly  understood,  requires  an  accurate  knowledge 
of  the  geography  of  the  seas  —  the  winds  and  currents  that  here 
and  there  prevail,  and  also  a  good  general  knowledge  of  astronomy. 

Running  a  line  in  surveying  and  running  a  course  at  sea,  are  math- 
ematically the  same  thing,  except  that  the  latter  is  on  a  larger  scale 
than  the  former,  without  its  accuracy,  and  it  is  for  a  different  object. 

In  surveying  we  take  no  account  of  the  magnitude  and  figure  of 
the  earth.  In  navigation  we  are  compelled  to  do  so,  unless  the 
limits  of  the  operation  be  very  small. 

There  are  two  methods  of  finding  the  position  of  a  ship. 

1st.  By  tracing  her  courses  and  distances,  as  in  a  survey.  This 
is  called  dead  reckoning. 

2d.  By  deducing  latitude  and  longitude  from  observations  on  the 
heavenly  bodies.  This  is  called  nautical  astronomy. 

(175) 


176  NAVIGATION. 

No  one  expects  accuracy  from  dead  reckoning,  and  as  a  general 
thing  it  is  only  used  from  day  to  day,  between  observations  ;  or  to 
keep  the  approximate  run  during  cloudy  weather  or  until  observa- 
tions can  again  give  a  new  starting  point. 

Some  navigators  keep  a  continuous  dead  reckoning  from  port  to 
port,  which  enables  them  to  judge  of  drifts,  currents,  and  unknown 
causes  of  error. 

The  earth  is  so  near  a  sphere  that  for  the  practical  purposes  of 
navigation  it  is  taken  as  precisely  so.  Its  magnitude  corresponds 
to  69  J  English  miles  to  one  degree  of  the  circumference,  but  in  the 
early  days  of  navigation  60  miles  were  supposed  to  be  about 
a  degree,  which  for  the  sake  of  convenience  is  still  retained. 

The  sixtieth  part  of  a  degree  is  called  a  nautical  mile,  and  it  is,  of 
course,  larger  than  an  English  mile. 

In  an  English  mile  there  are      -  5280  feet. 

In  a  nautical  mile  there  are  -     6079  feet. 

The  rate  which  a  ship  sails  is  measured  by  a  line  running  off  of 
a  reel,  called  the  log  line. 

The  log  is  nothing  more  than  a  piece  of  thin  board  in  the  form  of 
a  sector,  of  about  six  niches  radius,  the  circular  part  is  loaded  with 
lead  to  make  it  stand  perpendicular  in  the  water. 

The  line  is  so  attached  to  it  that  the  flat  side  of  the  log  is  kept 
toward  the  ship,  that  the  resistance  of  the  water  against  the  face  of 
the  log  may  prevent  it,  as  much  as  possible,  from  being  dragged 
after  the  ship  by  the  weight  of  the  line  or  the  friction  of  the  reel. 

The  time  which  is  usually  occupied  in  determining  a  ship's  rate 
is  half  a  minute,  and  the  experiment  for  the  purpose  is  generally 
made  at  the  end  of  every  hour,  but  in  common  merchantmen  at  the 
end  of  every  second  hour.  As  the  time  of  operating  is  half  a  min- 
ute, or  the  hundred  and  twentieth  part  of  an  hour,  if  the  line  were 
divided  into  120ths  of  a  nautical  mile,  whatever  number  of  those 
parts  a  ship  might  run  in  a  half  minute  she  would,  at  the  same  rate 
of  sailing,  run  exactly  a  like  number  of  miles  in  an  hour.  The  1 20th 
part  of  a  mile  is  by  seamen  called  a  knot,  and  the  knot  is  generally 
subdivided  into  smaller  parts,  called  fathoms.  Sometimes  (and  it  is 
the  most  convenient  method  of  division  )  the  knot  is  divided  into  ten 
parts,  more  frequently  perhaps  into  eight ;  but  in  either  case  the 
subdivision  is  called  a  fathom. 


INTRODUCTION.  177 

We  shall  consider  a  fathom  the  tenth  of  a  knot,  and  as  the  nauti- 
cal mile  is  6079  feet,  the  120th  part  of  it  is  60.66,  the  length  of  a 
knot  on  the  line,  and  a  little  over  5  feet  is  the  length  of  a  fathom. 

The  operation  of  ascertaining  the  rate  of  sailing  is  called  by  sea- 
men heaving  the  log. 

At  the  end  of  an  hour  the  loaded  chip,  or  log,  is  thrown  over  the 
stern  into  the  sea ;  a  quantity  of  the  line,  called  the  stray  line,  is 
allowed  to  run  off,  then  the  glass  is  turned  and  the  number  of  knots 
that  runs  off  the  reel  during  the  half  minute  is  the  rate  of  the  ship's 
motion. 

The  log  is  then  hauled  in  and  the  same  operation  is  repeated  at 
the  end  of  the  next  hour. 

The  officer  of  the  watch,  who  has  been  on  deck  during  the  hour, 
will  mark  on  the  slate  or  board,  called  the  log  board,  the  number 
of  miles  and  parts  of  a  mile  which  the  ship  has  sailed  during 
the  last  hour,  according  to  the  best  of  his  judgment ;  the  log  was 
thrown  only  to  help  make  up  that  judgment,  for  the  rate  at  the  time 
the  log  was  thrown  may  have  been  considerably  more  or  less  than 
the  average  motion  during  the  hour. 

The  course  of  a  ship  is  marked  by  the  mariner 's  compass. 

The  mariner's  compass  differs  from  the  surveyor's  compass  only 
by  its  construction,  that  is,  the  magnetic  needle  is  the  motive  power 
in  both.  In  consequence  of  the  motion  of  a  ship  at  sea,  the  mari- 
ner's compass  is  suspended  in  a  double  box,  moving  on  a  double 
axis,  one  at  right  angles  with  the  other,  the  whole  balanced  by  a 
central  weight  which  keeps  the  compass  card  nearly  steady  and  hori- 
zontal, whatever  be  the  motion  of  the  vessel. 

The  card  is  attached  to  the  needle,  and  is  moved  by  the  needle. 
The  card  is  divided  into  32  equal  parts,  called  points,  and  to  read 
over  these  points  in  order,  is  called  by  seamen,  boxing  the  compass, 
and  to  know  the  north  star  and  box  the  compass  is  too  often  the 
amount  of  the  common  sailor's  knowledge  of  navigation. 
12 


178  NAVIGATION. 

The  figure 
before  us  re- 
presents the 
card  of  the 
m  a  riner  *s 
compass.  The 
four  quadrant 
points  are 
marked  by  a 
single  letter  as 
W.  for  north, 
E.  for  east. 
The  midway- 
points  between 
these  by  two 
letters,  as  N. 
E.  for  north- 
east, N.  W.  for  north-west,  &c.  One  point  either  way  from  any  one 
of  these  eight  points  is  marked  by  the  word  by.  Thus,  N.  by  E.  is 
one  point  from  the  north  toward  the  east,  and  it  is  read  north  by 
east ;  S.  E.  by  S.  indicates  one  point  from  the  south-east  more  to 
the  south,  and  it  is  read  south-east  by  south  ;  W.  by  JV".  means  west 
one  point  toward  the  north. 

To  box  the  compass  we  begin  at  any  point,  as  north,  and  mention 
every  point  in  order  all  the  way  round,  thus  : 

North;  north  by  east;  north  north-east;  north-east  by  north;  north- 
east ;  north-east  by  east ;  east  north-east ;  east  by  north  ;  east,  &c. 

A  point  of  the  compass  is  11°  15',  which  is  subdivided  into  four 
equal  parts.  Mariners  never  take  into  account  a  less  angle  than  a 
quarter  point,  in  running  a  course. 

When  the  mariner  sets  his  course,  he  makes  allowance  for  the 
variation  of  the  needle,  and  his  magnetic  courses  he  reduces  to  true 
courses,  by  the  following 

RULE.  —  Make  a  representation  of  the  compass  card  on  paper,  and 
draw  a  line  through  the  compass  course. 

Now,  conceive  the  compass  card  turned  equal  to  the  variation  to  the 
eastward,  if  variation  is  west,  and  VICE  VERSA. 


INTRODUCTION.  179 

The  line  will  now  pass  over  the  true  course. 

In  the  following  examples,  the  true  courses  are  required. 

Answer. 
Compass  Course.  Variation.  True  Course. 

1.  S.  S.E.^E.  2  i  W.  S.  E.  b  E. 

2.  E.  \N.  3  E.  E.  S.  E.  |  S. 

3.  N.  W.  b  JF".  3£.#.  .V. bW.±W. 

4.  W.  S.  W.±S.  4  W.  S.  b  W.  i  W. 

5.  &  S.  W.    '  1 J  TF.  S.  £  PT. 

6.  jF.  bE.  N.E.\>E. 

7.  ^.  b  S.  2-fc  ^.  *S.  ^.  |  ^; 

8.  £.  60°#.  18°JF.  S.  78°#. 

9.  aV.  24  JF.  36  ^.  JV.  12  .#. 
10.  /S.  16  W.  40  .fir.  &  56  W. 

LEEWAY. 

The  angle  included  between  the  direction  of  the  fore  and  aft  line 
of  a  ship,  and  that  hi  which  she  moves  through  the  water,  is  called 
the  leeway. 

When  the  wind  is  on  the  right  hand  side  of  a  ship,  she  is  said  to 
be  on  the  starboard  tack ;  and  when  on  the  left  hand  side,  she  is  said 
to  be  on  the  larboard  tack  ;  and  when  she  sails  as  near  the  wind  as 
she  will  lie,  she  is  said  to  be  dose  hauled.  Few  large  vessels  will  lie 
within  less  than  six  points  to  the  wind,  though  small  ones  will  some- 
times lie  within  about  five  points,  or  even  less  ;  but,  under  such 
circumstances,  the  real  course  of  a  ship  is  seldom  precisely  in  the 
direction  of  her  head  ;  for  a  considerable  portion  of  the  force  of  the 
wind  is  then  exerted  in  driving  her  to  leeward,  and  hence  her  course 
through  the  water,  is  in  general  found  to  be  leeward  of  that  on  whicb 
she  is  steered  by  the  compass.  Therefore,  to  determine  the  point 
toward  which  a  ship  is  actually  moving,  the  leeway  must  be  allowed 
from  the  wind,  or  toward  the  right  of  her  apparent  course,  when  she 
is  on  the  larboard  tack ;  but  toward  the  left,  when  she  is  on  the  star- 
board tack. 

It  is  only  when  a  ship  crowds  to  the  wind,  that  leeway  is  made. 

It  is  seldom  that  two  ships  on  the  same  course  make  precisely  the 
same  leeway ;  and  it  not  unfrequently  happens,  that  the  same  ship 
makes  a  different  leeway  on  each  tack.  It  is  the  duty  of  the  officer 


180  NAVIGATION. 

of  the  watch,  to  exercise  his  best  skill  in  determining,  or  estimating, 
how  much  this  deviation  from  the  apparent  course  amounts  to  ;  and 
in  the  dark,  the  chief  reliance  must  be  placed  on  the  judgment  of  the 
experienced  mariner. 


CHAPTER    II. 

PLANE    SAILING. 

IN  plane  sailing,  the  earth  is  considered  as  a  plane,  the  meridians 
as  parallel  straight  lines,  and  the  parallels  of  latitude  as  lines  cut- 
ting the  meridians  at  right  angles.  And  though  it  is  not  strictly 
correct  to  consider  any  part  of  the  earth's  surface  as  a  plane,  yet 
when  the  operations  to  be  performed  are  confined  within  the  distance 
of  a  few  miles,  no  material  error  will  arise,  from  considering  them 
as  performed  on  a  plane  surface.  And,  as  we  have  already  seen,  in 
all  questions  where  the  nautical  distance,  difference  of  latitude,  de- 
parture, and  course,  are  the  objects  of  consideration,  the  results  will 
be  the  same,  whether  the  lines  are  considered  as  curves  drawn  on 
the  surface  of  the  globe,  or  as  equal  straight  lines  drawn  on  a  plane. 

In  all  maps,  and  charts,  and  constructions,  when  it  is  not  other- 
wise stated,  it  is  customary  to  consider  the  top  of  the  page  as  pointing 
toward  the  north,  the  lower  part  as  the  south,  the  right  side  as  the 
east,  and  the  left  as  the  west.  The  meridians  therefore,  in  any 
construction,  will  be  represented  by  vertical  lines,  and  the  parallels 
of  latitude,  by  horizontal  ones. 

Hence,  in  constructing  a  figure  for  the  solution  of  any  case  in 
plane  sailing,  the  difference  of  latitude  will  be  represented  by  a  ver- 
tical line,  the  departure  by  a  horizontal  one,  and  the  distance  by  the 
hypotenusal  line,  which  forms,  with  the  difference  of  latitude  and 
departure,  a  right-angled  triangle  ;  and  the  course  will  be  the  angle 
included  between  the  difference  of  latitude  and  distance. 

With  this  understanding,  the  solution  of  any  case  that  can  arise 
from  varying  the  data  in  plane  sailing  will  present  no  difficulty. 


PLANE    SAILING.  181 

EXAMPLES. 

If  a  ship  sail  from  Cape  St.  Vincent,  Portugal  (Lat.  37°  27  54" 
north),  S.  W.  \  S.  148  miles ;  required  her  latitude  in,  and  the 
departure  which  she  has  made  ? 

By  Construction.  —  Draw  the  vertical  line  AB,  to  represent 
the  meridian  ;  from  the  point  A,  make  the 
angle  BAC=3%  points,  the  given  course  ;  and 
from  a  scale  of  equal  parts,  take  A  C=  148  miles, 
the  given  distance  ;  from  C  on  AB,  draw  the 
perpendicular  CB,  then  AB  will  be  the  dif- 
ference of  latitude,  and  .ZZC'the  required  de- 
parture, and  measured  on  the  scale  from  which 
AC  was  taken,  AB  will  be  found  114.4,  and 
J3C93.9. 

Lat.  left        ...        37°  27  54"  N~ 
Diff.  lat.    -         -         -         -      1  54  24     N. 

Lat.  in  -        -        -        -        35°  8'  30'7  N.       Dep.  93.9 

2.  If  a  ship  sail  from  Oporto  Bar,  in  Lat.  41°  9'  north,  N.  W.  ±  W. 
315  miles  ;  required  her  departure  and  the  latitude  arrived  at  ? 

Ans.  Dep.  233.4  miles  TF.;  Lat.  44°  41'  JV. 

3.  If  a  ship  sail  from  lat.  55°  I7  N.,  S.  E.  by  S.,  till  her  depart- 
ure is  45  miles ;    required  the  distance  she   has   sailed  and  her 
latitude  ?  Ans.  Dis.  81  miles  ;  Lat.  53°  54'  N. 

4.  A  ship  from  lat.  36°  12'  N.,  sails  south-westward,  until  she 
arrives  at  lat.  35°  1'  N.,  having  made  76  miles  departure ;  required 
her  course  and  distance. 

Ans.  Course  S.  46°  57'  W. ;  Distance  104  miles. 

5.  If  a  ship  sail  from  Halifax,  in  lat.  44°  44'  N.,  S.  E.  %E.,  until 
her  departure  is  128  miles  ;  required  her  latitude  and  distance  sailed. 

Ans.  Lat.  42°  51'  N.,  and  dis.  sailed  165.6  miles. 

6.  A  ship  leaving  Charleston  light,  in  latitude  32°  43'  30"  north, 
sails  N.  eastward  128  miles,  and  is  then,  by  observation,  found  39 
miles  north  of  the  light ;  required  her  course,  latitude,  and  departure. 
Ans.  Lat.  33°  22'  30"  N. ;  Course  JV.  72°  16'  E.  ;  Dep.  122  miles. 

7.  A  ship  from  Cape  St.  Roque,  Brazil,  in  latitude  5°  107  south, 
sails  N.  E.  |  N.9  7  miles  an  hour,  from  3  P.  M.  until  10  A.  M.  ; 
required  her  distance,  departure,  and  latitude  in. 

Ans.  Dis.  133.  miles  ;  Dep.  84.4  miles  ;  Lat.  in,  3°  27'  south. 


182  NAVIGATION. 

8.  A  ship  from  latitude  41°  2'  JV7".,  sails  N.  N.  IF.  j  W.  5£  miles 
an  hour,  for  2£  days  ;  required  her  distance,  departure,  and  latitude 
arrived  at. 

Ans.  Dis.  330  miles  ;  Dep.  169.7  miles  ;  Lat.  45°  45'  N. 

Similar  examples,  might  be  given  without  end,  but  these  are 
sufficient,  for  they  only  involve  the  principles  of  the  solution  of  a 
plane  right-angled  triangle. 

In  the  preceding  examples  it  will  be  observed  that  we  traced  lat- 
itude from  latitude,  and  the  distances  east  and  west  we  called  de- 
parture —  not  difference  of  longitude.  It  now  remains  to  determine 
difference  of  longitude  from  departure. 

On  the  equator,  60  miles  of  departure  are  equal  to  one  degree  of 
longitude,  and  the  further  we  are  from  the  equator,  north  or  south, 
that  is,  the  greater  the  latitude  we  are  in,  the  same  departure  will 
cover  more  longitude. 

To  discover  the  law  for  changing  departure  to  difference  of  longi- 
tude, we  adduce  the  following  figure. 

Let  0  be  the  center  of  the  earth,  P  the 
pole,  P  C  a  portion  of  the  earth's  axis. 

The  plane  PCS  is  the  plane  of  one 
meridian,  and  P  CA  the  plane  of  another 
meridian. 

A  B  is  a  portion  of  the  equator  between 
the  two  meridians,  A  CB  is  a  sector  in 
the  plane  of  the  equator,  and  DEE  and 
FOQ  are  sectors  similar  to  A  CB. 

Observe  that  DE,  FG,  &c.,  are  parallels  of  latitude,  that  is,  they 
are  parallel  to  AB,  the  plane  of  the  equator. 

The  magnitude  of  DE  is  called  departure,  and  it  corresponds  to 
the  difference  of  longitude  AB. 

Also,  FG  is  departure  corresponding  to  the  same  difference  of 
longitude  AB.  The  difference  of  longitude  is  always  greater  than 
any  corresponding  departure,  that  is,  AB  is  obviously  greater  than 
any  other  parallel  distance  between  the  same  two  meridians. 


PLANE   SAILING.  183 

Because  the  two  sectors  A  CB  DEH  are  similar,  we  have  the 
proportion. 

AC  :  Dff  :  :  AB  :  DE  (1) 

Observe  that  A  C  is  the  radius  of  the  sphere,  Dff  is  the  sine  of 
the  arc  PD,  or  the  cosine  of  DA,  which  is  the  cosine  of  the  lat- 
itude of  the  point  D. 

Therefore  the  preceding  proportion  becomes, 

rad.  :  cos.  lat.  :  :  dif.  Ion.  :  dep. 
Or,         cos.  lat.  :  rad.  :  :  dep.  :  dif.  Ion. 

In  words,  The  cosine  of  the  latitude  is  to  the  radius  so  is  the  depar- 
ture to  the  difference  of  longitude. 

These  words  are  indelibly  engraved  on  the  memory  of  every  nav- 
igator, and  they  embrace  all  the  rules  that  can  be  made  for  chang- 
ing departure  into  longitude,  or  longitude  into  departure. 

When  a  ship  sails  east  or  west,  the  distance  sailed  is  called  de- 
parture, and  is  reduced  to  longitude  by  the  preceding  proportion. 
This  is  called  parallel  sailing. 

EXAMPLE  8. 

1.  What  difference  of  longitude  corresponds  to  47  miles  departure 
in  the  latitude  of  37°  23'  ?  Ans.  59.15  miles. 

Let  x=  the  difference  of  longitude  required. 

47  X  rad. 
Then        cos.  37°  53'  :  rad.  :  :  47  :  x=  cos  3?o  ^ 

By  log.  47  It     -  11.672098 

Cos.  37°  23  -       9.900144 

Diff.  Lon.  59.15  -  1.771954 

2.  How  many  miles,  or  how  much  departure  corresponds  to  a 
degree  in  longitude  on  the  parallel  of  42°  of  latitude  ? 

Ans.  44.59  miles. 
Here  the  longitude  of  one  degree  is  given, 

60  cos.  42° 
JR.  :  cos.  42°  :  :  60  :  x—  -^ 

By  log.  60  1.778151 

cos.  42°  ....      9.871073 


44.59      ....  1.649224 


PQ       ( 

CaPe  \ 


184  NAVIGATION. 

N.  B.  —  In  this  manner  the  length  of  a  degree  in  longitude  cor- 
responding to  each  degree  of  latitude  has  been  found  and  put  in  a 
table. 

3.  A  ship  sails  east  from  Cape  Race,  212  miles;  required  her  long- 
itude.    The  latitude  of  the  cape  is  43°  40'  N.,  longitude  53°  3'  15" 
west.  Ans.  Ion.  47°  54'  west. 

4.  Two  places  in  lat.  60°  12'  differ  in  longitude  34°  48';  required 
their  distance  asunder  in  miles.  Ans.  1336. 

5.  How  far  must  a  ship  sail  W.  from  the  Cape  of  Good  Hope,  that 
her  course  to  Jamestown,  St.  Helena,  may  be  due  north  ? 

Ans.  1  1  93  miles. 

lat.  34°  29'  S.  T  j  lat.  15°  55'  S. 

Ion.  18°  23'  K  Jamestown  j  ^  ^  43'  30"  W. 

6.  How  far  must  a  ship  sail  E.  from  Cape  Horn  to  reach  the  meri- 
dian of  the  Cape  of  Good  Hope  ?      The  latitude  of  Cape  Horn 
being  55°  58'  30"  S.,  Ion.  67°  21'  W.t  and  the  latitude  and  longi- 
tude of  the  Cape  of  Good  Hope  being  as  stated  hi  the  last  example. 

Ans.  2878  miles. 

7.  In  what  latitude  will  the  difference  of  longitude  be  three  times 
its  corresponding  departure  ?    In  other  words  in  what  latitude  will 
FG  be  one-third  of  AB  ?    (See  last  figure.) 

Ans.  lat.  70°  31'  44". 

8.  Take  the  2d  example  in  plane  sailing  (  page  181  ),  the  depar- 
ture made,  as  stated  in  the  answer,  is  233.4  miles.     What  is  the 
corresponding  difference  of  longitude  ?  Ans.  5°  18'  24". 

This  inquiry  now  arises.  To  what  latitude  does  this  departure 
correspond?  Is  it  to  the  latitude  left,  41°  9',  or  to  the  latitude  ar- 
rived at,  44°  41'  ?  Or  does  it  correspond  to  the  mean  latitude  be- 
tween the  two  ? 

If  we  suppose  the  departure  corresponds  to  lat.  41°  9',  then  the 
difference  of  longitude  by  the  preceding  rule  must  be  5°  10',  and  if 
the  departure  corresponds  to  lat.  44°  41',  then  the  difference  of  lon- 
gitude is  5°  28';  the  mean  of  these  is  5°  19',  and  if  we  take  the  de- 
parture to  correspond  with  the  mean  latitude  42°  55',  then  the  dif- 
ference of  longitude  would  be  5°  18'  24". 

In  the  examples  under  Plane  Sailing  we  have  supposed  the  earth 
a  plane,  and  the  course  a  ship  sails  a  straight  line,  but  neither  sup- 
position is  strictly  true. 


PLANE    SAILING.  185 

Meridians  are  wo£  parallel  with  each  other,  and  therefore  when  a 
ship  sails  by  the  compass,  and  cuts  all  the  meridians  at  the  same 
angle,  the  line  that  the  ship  sails  will  not  be  a  right  line  :  it  will  be 
a  curve  line  peculiar  to  itself,  called  a  rhumb  line. 

For  the  sake  of  illustration,  let  us  suppose 
that  in  the  annexed  figure,  P  is  the  north 
pole,  KQ  the  equator,  or  a  great  circle, 
every  part  of  which  is  a  quadrant  distance 
from  P  ;  PK,  PL,  PM,  &c.,  great  circles 
passing  through  P,  and  of  course  cutting 
the  equator  at  right  angles  ;  AI,  bB,  RS, 
<fec.,  arcs  of  smaller  circles  parallel  to  the 
equator,  and  therefore  cutting  the  meridians 
at  right  angles  ;  AE  a  curve  cutting  every 
meridian  which  it  meets,  as  PK,  PL,  PM,  &c.,  at  the  same  angle. 
Then  PK,  PL,  &c.,  produced  till  they  meet  at  the  opposite  pole, 
are  called  meridians  ;  AI,  bB,  RS,  <fec.,  continued  round  the  globe, 
are  called  parallels  of  latitude  ;  AE  is  called  the  rhumb  line,  passing 
through  A  and  E  ;  the  length  of  AE  is  called  the  nautical  distance 
from  A  to  E  ;  and  the  angle  BAb,  or  any  of  its  equals,  cBC,  dCD, 
<fcc.,  is  called  the  course  from  A  to  E. 

If  a  ship  sail  from  A  to  E,  EF  will  be  her  meridian  distance  ;  but 
if  she  sail  from  E  to  A,  ^4/will  be  her  meridian  distance. 

If  AB,  BC,  CD,  <fec.,  be  conceived  to  be  equal,  and  indefinitely 
small,  and  their  number  indefinitely  great  ;  then  the  triangles  ABb, 
BcC,  &c.,may  be  considered  as  indefinitely  small  right-angled  plane 
triangles.  And  as  the  angles  BAb,  CBc,  <fec.,  are  equal,  and  the 
right  angles  AbB,  BcC,  &c.,  are  equal,  the  remaining  angles  ABb, 
BCc,  &c.,  are  equal  ;  and  as  the  sides  AB,  BC,  &c.,  are  also  equal, 
these  elementary  triangles  ABb,  BCc,  CDd,  &c.,  will  be  all  iden- 
tical triangles  ;  therefore  AE  will  be  the  same  multiple  of  AB,  that 
the  sum  of  Ab,  Be,  Cd,  <fcc.,  is  of  Ab  ;  and  that  the  sum  of  Bb,  Cc, 
Dd,  &c.,  is  of  Bb. 

It  is  obvious  that 


the  whole  difference  of  latitude.     And, 

JBb+  Cfc-f-  X>d+JEt+&c.=t}ie  whole  departure. 


186  NAVIGATION. 

But  this  departure  is  neither  EF  nor  AI,  it  is  greater  than 
and  less  than  AI;  because  bB  is  less  than  its  corresponding  por- 
tion on  Al&nd.  greater  than  its  corresponding  portion  on  FE.  The 
same  may  be  said  of  cCt  Dd,  <fec.  Therefore  the  departure  on  any 
course  corresponds  to  neither  of  the  extreme  latitudes,  but  to  some  mean 
between  the  two,  and  it  is  so  near  the  arithmetical  mean,  that  the 
arithmetical  mean  is  taken  as  the  true. 

Therefore  hi  all  those  examples  in  Plane  Sailing,  on  page  181,  we 
can  take  the  departures  and  find  the  corresponding  differences  of 
longitude,  provided  we  take  the  middle  latitude  and  consider  the 
departure  run  on  that  parallel. 

This  method  of  connecting  the  change  in  longitude  with  a  ship's 
change  of  place  is  called 

MIDDLE     LATITUDE     SAILING. 

But  in  reality  there  is  no  such  thing  as  middle  latitude  sailing  ; 
the  cosine  of  the  middle  latitude  is  compared  to  the  radius,  as  the 
ratio  between  the  departure  made  and  the  corresponding  difference 
of  longitude,  but  the  departure  made  may  be  made  on  one  course  or 
on  several  courses.  When  a  ship  sails  on  several  courses  before 
the  run  is  summed  up,  the  summing  up  and  finding  the  result  in  one 
course  and  distance  is  called  working  a  traverse,  and  sailing  from  one 
point  to  another  by  several  courses  is  called 

TRAVERSE      SAILING. 

With  adverse  winds  or  crooked  channels,  vessels  are  obliged  to 
run  a  traverse.  Going  round  a  survey  and  keeping  an  account  of 
our  course  and  distance  from  the  starting  point  is  working  a  traverse, 
and  the  operation  is  the  same  on  sea  or  land,  except  on  land  we  aim 
at  coming  round  to  the  same  point  again,  but  on  sea  we  wish  to  make 
some  other  point. 

With  this  explanation  it  is  obvious  that  we  must  make  a  table  as 
in  a  survey,  and  compute  the  course  and  distance  from  the  starting 
point,  and  this  is  called  the  course  and  distance  made  good. 

To  work  a  traverse  we  use  the  traverse  table  of  course  ;  that  table 
is  made  to  every  half  degree,  and  the  column  in  the  table  nearest  to 
the  course  is  sufficiently  exact. 


TRAVERSE   SAILING. 


187 


The  following  table  gives  the  degree  and  parts  of  a  degree  corres- 
ponding to  every  point  and  quarter  point  of  the  compass. 

Deg. 

47°  48'  45" 
50°  37'  30" 
53°  26'  15" 
56°  15' 
59°  3' 45" 
61°  62' 30" 
64°  41'  15" 
67°  30' 
70°  18'  45" 
73°  7'  30" 
75°  56'  15' 
78°  45' 
81°  33'  45" 
84°  22'  30" 
87°  11'  15" 
90°  0' 

In  works  exclusively  designed  for  practical  navigation,  the  traverse 
table  is  adapted  exactly  to  the  points  and  quarter  points  of  the 
compass,  but  the  table  hi  this  work  is  sufficient  for  the  purpose. 

The  use  of  this  table  is  to  find  the  degree  corresponding  to  any 
given  course,  thus  :  JV.  by  E.t  N,  by  W.,  S.  by  E.,  S.  by  W.t 
each  correspond  to  1  point  or  11°  15'.  In  using  our  traverse  table 
for  1  point  we  should  take  a  mean  result  between  11°  and  11°  30', 
which  mean  result  can  be  taken  by  the  eye. 

Again  S.E.  by  E.  £  E.  is  5£  points,  or  S.  59°  E.  nearly,  and  so  on 
for  any  other  course  that  may  be  named. 

The  student  is  now  fully  prepared  to  work  the  following  examples 
in  traverse  sailing. 

EXAMPLES. 

1.  A  ship  from  Cape  Clear,  Ireland,  in  lat.  51°  25'  N.  and  longitude 
9°  29'  W.,  sails  as  follows  : 

S.  S.E.$E.\S  miles,  E.  S.  E.  23  miles,  S.  W.  by  W.  %  W.  36 
miles,  W.  £  N.  12  miles,  and  S.  E.  by  E.  £  E.  41  miles. 


Point*. 

Deg. 

Point 

i 

2°  48'  45" 

4-J 

i 

5°  37'  30" 

4£ 

S 

8°  26'  15" 

4| 

1 

11°  15' 

5 

1£ 

14°  3'  45" 

5£ 

H 

16°  52'  30" 

% 

i| 

19°  41'  15" 

5J 

2 

22°  30' 

6 

2i 

25°  18'  45" 

6i 

^ 

28°  r  so'- 

6* 

2| 

30°  56'  15" 

6| 

3 

33°  45' 

7 

3£ 

36°  33'  45" 

7i 

3£ 

39°  22'  30" 

H 

I 

42°  11'  15" 

7| 

45°  0' 

8 

188 


NAVIGATION. 


Required  her  course  and  distance  made  good,  the  departure,  lati- 
tude and  longitude  of  the  ship. 

By  Construction.  —  Take  A  for  the  place  sailed  from,  and  draw 
the  vertical  line  NASC,  to  represent  the  meridian.  About  A,  as  a 
center,  with  the  chord  of  60°, 
describe  a,  circle,  cutting  NO  in 
JFand  S;  then  ^and  5  will 
represent  the  north  and  south 
points  of  the  compass.  Take  2J 
points  from  the  line  of  chords,* 
and  apply  it  from  S  to  a,  join 
Aa,  and  on  it  take  AD=16  from 
a  line  of  equal  parts.  Then  D 
will  be  the  place  of  the  ship  at 
the  end  of  the  first  course.  From 
St  set  off  £6=6  points  from  the 
line  of  chords  ;  join  A  b,  and 
through  D  draw  DE,  parallel  to 
Ab,  and  make  it  equal  to  23  from 
the  same  scale  of  equal  parts 
that  AD  was  taken  from.  Then 
E  will  be  the  place  of  the  ship, 
at  the  end  of  the  second  course.  Make  Sc—5^  points,  Nd  7£ 
points,  and  Se  5^  points,  taken  from  the  line  of  chords.  Through 
E  draw  EF,  parallel  to  Ac,  and  make  it  equal  to  36,  from  the  scale 
of  equal  parts ;  through  F  draw  FQ  parallel  to  Ad,  and  make  it 
equal  to  12,  from  the  scale  of  equal  parts  ;  through  G  draw  GB 
parallel  to  Ae,  and  make  it  equal  to  41,  from  the  scale  of  equal 
parts.  Demit^?  C  a  perpendicular,  on  the  meridian  NC,  and  join 
AB  ;  then  B  will  be  the  place  of  the  ship,  AB  her  distance  from 
the  place  which  she  left,  A  C  her  difference  of  latitude,  BC  her  de- 
parture, and  BA  C  the  course  which  she  has  made  on  the  whole. 
Now,  AB,  AC,  and  BC  being  measured  on  the  scale  of  equal  parts 
from  which  the  distances  were  taken,  we  have  AB=6%.7,  AC= 
59.6,  and  .5(7=19.6  miles.  And  the  arc  included  by  A  C  and  AB 
if  measured  on  the  line  of  chords,  gives  about  1 8°  for  the  measure 
of  the  course  BA  C. 


»  Two  and  J  points  is  25°  18'  45",  that  is,  take  the  chord  of  25°  18'  in  the 
dividers,  and  set  it  off  from  s  to  a,  and  so  on,  for  other  angles. 


TRAVERSE   SAILING. 

TRAVERSE     TABLE. 


169 


Course*. 

Points. 

Vis.  I 

Diff.  Lat 

Deo. 

N. 

s. 

E.      1 

w. 

8.  S.E.IK 

E.  S.  E. 

? 

16 
23 

14.5 
8.8 

6.8 
21.3 

S.  IT.  by  W.\W. 

6i 

36 

17.9 

31.8 

W.^N. 

7£ 

12 

1.8 

11.9 

S.  E.  by  E.  ±E. 

6i 

41 

21.1 

35.2 

1.8 

61.4 

63.3 

45.7 

1.8 

43.7 

Result 


59.6  |  19.6 

Lat.  left        -        -       51°  25'    N. 
diff.  lat.    -        -         -     1    00     S. 

Lat.  in        -        -        50   25    N.    Mid.  lat.  50°  55' 
To  find  the  course  and  distance,  by  trigonometry, 
As  dif.  lat.  59.6* miles  1.775246 

:  radius  90°  10.000000 

:  :  dep.  19.6  miles  1.292256 

:  tan.  course  18°  12' 

As  sin.  course  18°  12' 
:  dep.  19.6  miles 
:  Radius 

:  Dis.  62.75  miles 
To  find  the  dif.  of  longitude. 
As  cos.  50°  55' 

:  Radius 
: :  dep.  19.6 
:  diff.  Ion.  31.09  miles 

Longitude  left 
diff.  Ion. 
Lon.  in 


9.517010 

9.484621 

1.292256 

10.000000 

1.797635 

9.799651 

10.000000 

1.292256 


1.492605 

9°  29'  west 

31    east 

IF'SS7  west 


Thus,  we  have  found  the  course  18°  12' ;  Distance  62.75  miles  ; 
diff.  longitude  31'  E.\  lat.  in  50.25  N.\  Ion.  8°  58'  W. 

If  these  be  the  distances  run  hi  a  day,  from  noon  to  noon  again, 
then  the  preceding  operation  is  called  working  a  day's  work  ;  other- 
wise it  is  called  working  a  traverse,  as  we  have  mentioned  before. 


190  NAVIGATION. 

But  this  is  not  the  seaman's  way  of  working  a  day's  work,  he  does 
it  all  by  inspection,  in  the  traverse  table.  For  example,  taking  the 
result  of  the  traverse  59.6  south,  and  19.6  east,  which  shows  that 
the  resulting  course  is  between  the  south  and  the  east,  and  with 
these  numbers  he  enters  the  traverse  table,  and  finds,  as  near  as 
possible,  59.6  and  19.6,  standing  as  latitude  and  departure  ;  and 
they  are  found  nearly  under  the  angle  of  18°,  and  opposite  the  dis- 
tance 63.  nearly. 

Hence,  he  takes  his  course  as  S.  18°  E.,  and  dis.  63.  To  find 
the  difference  of  longitude,  he  takes  the  middle  latitude  as  a  course, 
and  the  departure  as  difference  of  latitude,  then  the  distance  in  the 
table  is  difference  of  longitude. 

In  this  instance,  we  take  51°  as  a  course,  and  in  the  difference  of 
latitude  column  we  find  19.5,  and  the  distance  opposite  to  it  is  31., 
which  we  take  for  difference  of  longitude. 

The  reason  for  this  is  as  follows  : 

For  the  longitude  we  have, 

cos.  mid.  L  :  R  : :  dep.  :  diff.  Ion.  ( 1 ) 

In  the  construction  of  the  traverse  table,  we  have, 

cos.  course  :  JR.  :  :  diff.  lat.  :  dist.  (2) 

Now,  in  proportion  (2),  if  we  take  the  middle  latitude  for  a  course, 
and  the  dep.  for  difference  of  latitude  ;  it  necessarily  follows,  th  at 
the  last  term  of  proportion  (2)  must  be  diff.  of  longitude ;  for 
proportion  (2)  would  then  be  transformed  into  proportion  (1). 

2.  A  ship  sails  from  Cape  Clear,  as  follows  ;  /S.  by  W.  23  miles  ; 
W.  S.  W.  40  miles  ;  S.  W.  £  W.  18  miles  ;  W.  %  N.  28  miles  ;  S.  by 
E.  12  miles  ;  S.  S.  E.  j  E.  1 6  miles. 

Required  the  course  and  distance  made  good,  and  the  latitude 
and  longitude  arrived  at. 

Ans.  Course  S.  45°  47'  W. ;  dis.  102.4  miles. 
Latitude  of  ship  50°  14'  JV. ;  Lon.  11°  25'  W. 

3.  A  ship  at  noon,  on  a  certain  day,  was  hi  lat.  41°  12'  N.,  and 
longitude  37°  21'  W.,  she  then  sailed  as  follows : 

S.  W.  by  W.  21  m.  ;  S.  W.  $  S.  31  m.  ;  W.  S.  W.  4  S.  16  m.  ; 
S.  %E.  18m.;  S.  W.  $  W.  14,  and  W.  %  N.  30  miles. 

Required  her  course,  distance,  latitude,  and  longitude. 
Ans.  course  S.  52°  49'  W.;  dis.  111.7  ;  lat. 40°  5'  N.;  Ion.  39°  18' TF. 


SAILING    IN    CURRENTS.  191 

4.  Last  noon  we  were  in  latitude  28°  46'  south,  and  longitude 
32°  20'  west ;  since  then  we  have  sailed  by  the  log  : 

8.  JF.j  W.  62  m.  ;  S.  by  Fl  16  m. ;  W.  \  S.  40  m. ;  S.W.  |TF". 
29  m. ;  S.  byK  30  m. ;  and  S.  j  E.  14  miles.    Required  the  direct 
course  and  distance,  and  our  present  latitude  and  longitude. 
Ans.  Course  S.  43°  14'  W. ;  Dis.  158  m. ;  lat.  30°  41'  S.; 

Ion.  34°  24'  W. 

5.  A  ship  from  Toulon,  lat.  43°  7'  N.9  Ion.  5°  56,  K,  sailed 

5.  S.  W.  48  m. ;  S.  by  E.  34  m. ;  8.  W.  J  W.  26  m. ;  and  E. 
17  miles.     Required  her  course  and  distance  to  Port  Mahon,  Lat. 
39°  52'  N.,  and  longitude  4°  18  30"  east. 

Ans.  Lat.  of  ship  41°  32'  N. ;  Ion.  5°  37'  east. 
Course  to  Port  M.  S.  31°   W.  nearly,  and  distance  117.5  miles. 

6.  On  leaving  the  Cape  of  Good  Hope,  for  St.  Helena,  we  took 
our  departure  from  Cape  Town,  bearing  S.  -E.by  S.  12  miles,  after 
running  N.  W.  36  m.,  and  N.  W.  by  W.  140  miles.     Required  our 
latitude  and  longitude,  and  the  course  and  distance  made. 

N.  B.  Lat.  of  Cape  Town  33°  56'  S.          Lon.  18°  23'  JE. 

Lat.  of  St.  Helena    15°  55'  S.          Lon.     5°  43'  30"    W. 
Ans.  Lat.  32°  3'  S. ;  Lon.  15°  25'  E. ;  course  N.  52°  41'  W. ; 

dis.  187  miles. 

SAILING     IN     CURRENTS. 

If  a  ship  at  B,  sailing  in  the  direction  BA,  were  in  a  current  which 
would  carry  her  from  B  to  C,  in  the  same  tune  that  in  still  water  she 
would  sail  from  B  to  A,  then,  by  the  joint 
action  of  the  current  and  the  wind,  she  would 
in  the  same  time,  describe  the  diagonal  BD 
of  the  parallelogram  AB  CD.  For  her  being 
carried  by  the  current  in  a  direction  parallel 
to  BC,  would  neither  alter  the  force  of  the 
wind,  nor  the  position  of  the  ship,northe  sails,  with  respect  to  it ;  the 
wind  would  therefore  continue  to  propel  the  ship  in  a  direction 
parallel  to  AB,  in  the  manner  as  if  the  current  had  no  existence. 
Hence,  as  she  would  be  swept  to  the  line  CD,  by  the  independent 
action  of  the  current,  in  the  same  time  that  she  reached  the  line  AD, 
by  the  independent  action  of  the  wind  on  her  sails,  she  would  be 
found  at  D,  the  point  of  intersection  of  the  lines  AD  and  CD,  hav- 
ing moved  along  the  diagonal  BD. 


192  NAVIGATION. 

Now  the  log  heaved  from  the  ship  in  the  ordinary  way,  can  give 
no  imitation  of  a  current ;  for  the  line  withdrawn  from  the  reel  is 
only  the  measure  of  what  the  ship  sails  from  the  log ;  and,  conse- 
quently, as  the  log  itself,  as  well  as  the  ship,  will  move  with  the 
current,  the  distance  shown  by  the  log  in  a  current,  is  merely  what 
it  would  have  been  if  the  ship  had  been  in  still  water. 

If  the  ship  sail  in  the  direction  of  the  current,  the  whole  effect  of 
the  current  will  be  to  increase  the  distance  ;  but  if  she  sail  against 
the  current,  the  difference  between  the  rate  of  sailing  given  by  the 
log  and  drift  of  the  current,  will  be  the  distance  which  the  ship 
actually  goes  ;  and  she  will  move  forward,  if  her  rate  of  sailing  be 
greater  than  the  drift  of  the  current,  but  otherwise,  her  motion  will 
be  retrograde,  or  she  will  be  carried  backwards,  in  the  direction  of 
the  current. 

Problems  relating  to  the  oblique  action  of  a  current  upon  a  ship, 
may  be  resolved  by  the  solution  of  an  oblique-angled  plane  triangle, 
such  as  ABD,  in  the  preceding  figure,  where  if  AB  represent  the 
distance  which  a  ship  would  sail  in  still  water,  and  AD  the  drift  of 
the  current  in  the  same  time,  BD  will  be  the  actual  distance  sailed, 
and  ABD  the  change  in  the  course  produced  by  the  current. 

A  great  variety  of  problems  might  be  proposed  relative  to  currents, 
but  the  chief  ones  of  any  practical  importance,  are  the  following : 

1.  To  determine  a  ship's  actual  course  and  distance  in  a  current, 
when  her  course  and  distance  by  the  compass  and  the  log,  and  the 
setting  and  drift  of  the  current,  are  given. 

2.  To  find  the  course  to  be  steered  through  a  known  current,  the 
required  course  in  still  water,  and  the  ship's  rate  of  sailing,  being 
known. 

3.  To  find  the  setting  and  drift  of  a  current,  from  a  ship's  actual 
place,  compared  with  that  deduced  from  the  compass  and  the  log. 

The  first  of  these  cases  may  be  conveniently  resolved,  by  con- 
sidering the  ship  as  having  performed  a  traverse,  the  setting  and 
drift  of  the  current  being  taken  as  a  separate  course  and  distance. 

EXAMPLES. 

1.  If  a  ship  sail  W.  28  miles  in  a  current,  which  in  the  same  time 
carries  her  N.  N.  W.  8  miles,  required  her  true  course  and  distaace. 


SAILING    IN    CURRENTS. 


193 


N.  B.  Conceive  the  current  to  be  one  course  and  distance,  and 
with  the  other  courses  find  the  course  and  distance  made  good. 
Thus,  by  the  traverse  table  : 


Course. 

Vis. 

Diff 

N. 

.  Lat 
8. 

D« 
E. 

p'  w. 

w. 

N.  N.  W. 

28 
8 

7.39 

28 
3.06 

7.39 

31.06 

As  7.39  :  rad.  :  :  31.06  :  tan.  76°  36',  the  course, 
cos.  76°  31'  :  R  :  :  31.06  :  31.93,  the  distance. 

2.  If  a  ship  sail  E.  7  miles  an  hour  by  the  log,  in  a  current  setting 
E.  N.  E.  2.5  miles  per  hour ;  required  her  true  course,  and  hourly 
rate  of  sailing. 

Ans.  Course  N.  84°  8'  E.,  and  rate  9.358  per  hour. 

3.  A  ship  has  made  by  the  reckoning  N.  -J  W.  20  miles,  but  by 
observation  it  is  found,  that,  owing  to  a  current,  she  has  actually 
gone  N.  N.  E.  28  miles.     Required  the  setting  and  drift  of  the  cur- 
rent in  the  time  which  the  ship  has  been  running. 

Ans.  Setting  N.  64°  48'  E.,  and  drift  14.1  miles. 

4.  A  ship's  course  to  her  port  is  W.  N.  JF.,  and  she  is  running 
by  the  log  8  miles  an  hour,  but  meeting  with  a  current  setting 

W.  -J-  S.  4  miles  an  hour,  what  course  must  she  steer  in  the  current 
that  her  true  course  may  be  W.  N.  W  ? 

Ans.  Course  N.  44°  39'  W. 

5.  In  a  tide  running  N.  W.I)  W.  3  miles  an  hour,  I  wished  to 
weather  a  point  of  land,  which  bore  N.  E.  14  miles.     What  course 
must  I  steer  so  as  to  clear  the  point,  the  ship  sailing  7  miles  an  hour 
by  the  log,  and  what  time  shall  I  be  in  reaching  the  point  ? 

Ans.  Course  N.  69°  51'  JE.,  and  time  2  hours  25  minutes. 

6.  From  a  ship  hi  a  current,  steering  W.  S.  W.  6  miles  an  hour 
by  the  log,  a  rock  was  seen  at  6  in  the  evening,  bearing  S.  W.  ^  S. 
20  miles.    The  ship  was  lost  on  the  rock  at  1 1  P.  M.    Required  the 
setting  and  drift  of  the  current. 

Ans.  Setting  S.  75°  10'  j&,  and  drift  3.11  miles  per.  hour. 
13 


194  NAVIGATION. 

C  HA  P  T  E  R     III. 

MERCATOR'S    CHART    AND    MERCATOR'S 
SAILING. 

IN  representing  any  small  portion  of  the  earth's  surface,  it  is  suf- 
ficiently accurate  to  represent  the  meridians  as  parallel ;  but  if  the 
portion  of  the  earth  is  considerable,  the  representation  will  not  be 
true  unless  the  meridians  are  curved. 

If  we  make  a  chart  and  draw  all  the  meridians  parallel  with  each 
other,  the  length  of  a  degree  of  longitude  in  all  places,  except  on  the 
equator,  will  be  greater  on  the  chart  than  its  true  distance,  but  the 
true  bearing  of  one  place  from  another  will  be  preserved,  provided 
we  increase  the  degrees  of  latitude  in  the  same  ratio  as  the  degrees 
of  longitude  are  increased. 

Gerrard  Mercator,  a  Fleming,  in  1556,  published  a  chart  which 
seemed  to  embrace  this  idea,  but  he  did  not  show  its  construction, 
nor  were  his  degrees  in  their  true  proportion  ;  but  from  this  came 
the  name  of  Mercator's  Chart. 

A  Mr.  Wright,  an  Englishman,  in  1599,  it  is  said,  published  the 
true  sea  chart,  constructed  on  the  following  principles. 

1.  The  distance  between  two  meridians  at  the  equator,  into  their  dis- 
tance in  any  parallel  of  latitude,  as  the  radius  is  to  the  cosine  of  that 
latitude. 

2.  Any  part  of  a  parallel  of  latitude,  is  to  a  like  part  of  the  meri- 
dian, as  the  radius  is  to  the  secant  of  that  parallel. 

We  shall  make  an  effort  to  illustrate  these  principles  by  the  fol- 
lowing figure. 

Conceive  the  equator  to  be  extended  both  ways  parallel  to  the 
earth's  axis,  thus  forming  a  cylinder,  whose  circumference  is  just 
equal  to  the  circumference  of  the  earth. 


MERCATOR'S    CHART. 


195 


Let  Qg  be  the  plane  of  the  equator,  Pp  the  earth's  axis  ;  con- 
ceive a  globe  enclosed  in  the  cylinder,  HLNM. 

Suppose  there  is  an  island  on  the  earth  at  a,  that  island  is  projected 
on  the  cylinder  at  A.  The  surface  of  the  earth  at  b  is  projected  at 
J5.  Conceive  this  paper  cylinder  cut  by  a  line  at  right  angles  to 
the  equator  and  rolled  out,  it  will  then  be  a  true  representation  of 
Mercator's  chart. 

The  scale  on  the  globe  at  a  is,  to  the  scale  on  the  chart  at  A,  as 
Ca  to  CAt  that  is,  as  radius  to  the  secant  of  the  latitude  at  a. 

The  scale  on  the  chart  at  A  is,  to  the  scale  on  the  chart  at  B,  as 
CA  is  to  CB,  that  is,  the  scale  on  the  chart  increases  as  the  secants 
of  the  latitudes  increase. 

The  poles  of  the  earth,  and  places  very  near  the  poles,  can  never 
be  represented  on  this  chart. 

The  meridian  distance  of  a  degree  on  the  globe,  as  at  a,  is  60 
miles,  on  the  chart  at  A  it  is  60,  into  A  0  the  secant  of  the  latitude, 
calling  Ca  unity. 

If  we  commence  at  the  equator  at  Q,  and  take  one  mile  for 
unity.  Then, 

Mer.  pts.  of  1'=  nat.  sec.  1 
Mer.  pts.  of  2'=  nat.  sec.  1-|-  nat.  sec.  2 
Mer.  pts.  of  3'=  nat.  sec.  1'+  nat.  sec.  2'+  nat.  sec.  3 
Mer.  pts.  of  4'=  nat.  sec.  I'-f-  nat.  sec.  2'-f-  nat.  sec.  3' 
+  nat.  sec.  4',  <fec.,  &c. 

In  this  manner  the  table  of  meridional  parts  was  originally  con- 
structed. It  is  Table  IV  of  this  work. 

The  following  figures  represent  any  problem  than  can  arise  in 
Mercator's  sailing. 

A  0  represents  the  true  dif- 
ference of  latitude. 

-4Z>  represents  the  meridion- 
al difference  of  latitude,  which 
is  always  taken  from  the  table. 

CB  represents  the  departure. 

DE  the  difference  of  longi- 
tude. 


196  NAVIGATION. 

AS  represents  the  distance. 
A,  the  angle  at  A,  represents  the  course. 
Three  of  these  six  quantities  must  be  given  to  solve  a  problem. 
Observe  that  the  difference  of  longitude  DE  is  always  greater 
than  the  departure  CB,  as  it  ought  to  be. 

EXAMPLES. 

1.  A  ship  from  Cape  Finisterre,  in  lat.  42°  56'  N.t  and  longitude 
8°  16'  W.,  sailed  S.  W.  J  W.  till  her  difference  of  longitude  is  134 
miles  ;  required  the  distance  sailed  and  the  latitude  in. 

By  logarithms.     As  radius 10.000000 

:  diff.  Ion.  134  miles  -  -  -  -  2.127105 
:  :  cot.  course  4£  points  -  -  -  9.957295 
:  mer.  diff.  lat.  121.5  miles  -  -  2.084400 

Lat.  Cape  Finisterre  42°  56'  N.  Mer.  parts   -     2858 

Mer.  diff.       -     121 

Lat.  41°  27'  N.9  corresponding  to       -     -     -     2737  in  table 

As  cosine  course 9.827085 

:  proper  diff.  lat.  89  miles      -     -  1.949390 
:  :  radius 10.000000 


:  dis.  132.5  miles    ....    -2.122305 

2.  A  ship  from  lat.  40°  41'  N.,  Ion.  16°  37'  W.,  sails  in  the  JV. 
E.  quarter  till  she  arrives  hi  lat.  43°  57'  N.t  and  has  made  248  miles 
departure ;  required  her  course,  distance,  and  longitude  hi. 

Ans.  course  N.  51°  41'  E.y  dis.  316  miles,  and  Ion.  in  11°  W. 

3.  How  far  must  a  ship  sail  N.  E.  \  E.  from  lat.  44°  12'  N.,  Ion. 
23°  W.,  to  reach  the  parallel  of  47°  N.,  and  what  from  that  point 
will  be  the  bearing  and  distance  of  Ushant,  which  is  in  lat.  48°  28' 
N.  and  Ion.  5°  3'  W."* 

Ans.  She  must  sail  262  miles,  and  her  course  and  distance  to 
Ushant  will  then  be  N.  80°  32'  E.,  and  dis.  535  miles. 

4.  A  ship  from  the  Cape  of  Good  Hope  steers  E.  %  S.  446  miles, 
required  her  place,  and  her  course,  and  distance  to  Kerguelen'f 
Land,  in  lat.  48°  41'  S.,  and  Ion.  69°  cast. 


MECATOR'S    SAILNIG.  197 

Ans.  lat.  in  35°  13'  S.,  Ion.  in  27°  21'  E.,  course  S.  66°  25'  £., 
and  distance  20 IS  miles. 

5.  By  observation,  a  ship  was  found  to  be  in  lat.  41°  50'  S.,  Ion. 
68°  14  E.  She  then  sailed  N.  E.  140m,  and  E  %  S.  76m;  required 
her  place,  and  her  course,  and  distance  to  the  island  of  St.  Paul, 
which  is  in  lat.  38°  42'  S.t  and  in  Ion.  77°  18'  E. 

Ans.  lat  40°  18'  S.,  Ion.  72°  2',  course  N.  68°  35'  E,  and  din. 
263  miles,  nearly. 


CELESTIAL  OBSERVATIONS. 


CHAPTER    I. 

WE  now  come  to  the  more  scientific  and  essential  parts  of  navi- 
gation, the  determination  of  latitude  and  longitude  .by  celestial 
observations. 

We  shall  at  present  confine  ourselves  to  latitude,  first  calling  to 
mind  the  following  necessary  definitions  and  explanations  : 

1.  MERIDIAN.  —  The  meridian  of  any  place  is  the  north  and  south 
line  passing  through  that  place,  and  it  may  be  conceived  to  run  along 
the  ground  or  pass  in  the  same  direction  in  the  heavens,  through  the 
point  vertically  over  the  place.     The  line  on  the  earth  is  the  terres- 
trial meridian,  the  line  in  the  heavens  is  called  the  celestial  meridian  ; 
they  are  both  in  one  plane  with  the  center  of  the  earth. 

2.  EQUATOR.  —  The  equator  is  that  circle  around  the  earth  over 
which  the  sun  seems  to  pass  when  the  days  and  nights  are  equal 
all  over  the  earth. 

3.  LATITUDE.  —  The  latitude  of  any  place  is  the  meridian  distance 
of  that  place  from  the  equator,  measured  by  degrees  and  parts  of  a 
degree  of  arc. 

4.  LONGITUDE.  —  The  longitude  of  any  place  is  the  inclination  of 
the   plane  of  its  meridian,  with  the  plane  of  some  other  definite 
meridian  from  which  reckoning  is  made.     This  inclination  is  meas- 
ured on  the  equator  by  degrees,  minutes,  and  seconds  of  arc,  and  it 
is  either  east  or  west.* 

*  The  first  meridian  to  reckon  from  may  be  arbitrarily  chosen,  and  different 
nations  have  taken  different  meridians  for  the  commencement  of  longitude,  but 
custom  and  long  association  have  pretty  firmly  fixed  the  meridian  of  Greenwich 
(England)  as  the  first  meridian  for  all  who  use  the  English  language. 
(198) 


NAVIGATION.  199 

5.  DECLINATION.  —  The  declination  of  a  heavenly  body  is  its 
meridian  distance  from  the  equator  north  or  south. 

6.  POLAR  DISTANCE.  —  The  polar  distance  of  a  body  is  its  decli- 
nation added  to,  or  subtracted  from  90°.     If  both  added  and  sub- 
tracted, we  shall  have  the  meridian  distances  from  each  pole. 

The  distance  from  the  north  pole,  is  called  north  polar  distance, 
and  from  the  south  pole,  south  polar  distance.  The  two  polar  dis- 
tances must  of  course  make  1 80°. 

7.  ZENITH.  —  Zenith  is  the  point  in  the  heavens  directly  over- 
head. 

8.  HORIZON.  —  The  horizon  is  either  apparent  or  real,  or  as  com- 
monly expressed,  sensible  or  rational. 

The  sensible  horizon  is  a  plane  conceived  to  touch  the  earth  at  any 
point  at  which  an  observer  is  situated. 

The  rational  horizon  is  a  plane  parallel  to  the  sensible  one,  passing 
through  the  center  of  the  earth. 

The  zenith  is  the  pole  to  the  horizon. 

9.  GREAT  CIRCLES.  —  A  great  circle  in  the  heavens  is  any  circle 
whose  plane  passes  through  the  center  of  the  earth. 

All  great  circles  which  pass  through  the  zenith  are  perpendicular 
to  the  horizon,  and  such  circles  are  called  vertical  circles,  azimuth 
circles,  or  circles  of  altitude. 

10.  AZIMUTH.  —  The  angle  which  the  meridian  makes  with  that 
vertical  circle  which  passes  through  any  object  is  said  to  be  the 
azimuth  of  that  object.     Hence,  azimuths  may  be  reckoned  from  the 
north  or  south  points  of  the  horizon. 

11.  ALTITUDE.  —  The  altitude  of  any  object  is  its  angular  distance 
from  the  horizon,  measured  on  a  vertical  circle.* 

Altitudes  are  very  frequently  measured  at  sea,  several  times  in  a 
day  hi  fair  weather  ;  but  altitudes  observed  from  the  surface  of  the 
earth,  or  above  it,  require  several  corrections  before  the  true  alti- 
tudes can  be  deduced  from  them. 

*  We  do  not  pretend  to  give  all  the  definitions  of  the  sphere,  but  we  suppose 
the  reader  is  already  acquainted  with  them,  from  his  knowledge  of  Geography 
and  Astronomy. 


200  CELESTIAL   OBSERVATIONS. 

These  corrections  are  for  semi-diameter,  dip,  refraction,  and  parallax. 
The  correction  for  semi-diameter  is  obvious. 

At  sea,  the  visible  horizon  (from  which  all  observed  altitudes  are 
taken)  is  where  the  sea  and  sky  apparently  meet,  and  when  the  eye 
of  the  observer  is  above  the  water,  this  visible  horizon  is  below  the 
sensible  horizon,  and  the  amount  of  the  depression  is  called  the  dip 
fthe  horizon.  Its  correction  is  always  subtractive,  and  its  amount 
is  to  be  found  in  Table  V. 

Refraction  is  to  be  found  in  Table  VII.  It  is  always  subtractive, 
and  for  the  reason,  see  some  treatise  on  natural  philosophy. 

Parallax  is  always  additive.  Conceive  two  lines  drawn  to  a  hea- 
venly body ;  one  from  an  observer  at  the  circumference  of  the  earth 
and  the  other  from  the  center  of  the  earth,  the  inclination  of  these 
two  lines  is  parallax,  and  when  the  body  is  in  the  horizon  its  parallax 
is  greatest,  and  it  is  then  called  horizontal  parallax. 

Parallax  always  tends  to  depress  the  object,  but  the  parallax  of 
any  celestial  object,  except  that  of  the  moon,  is  so  small,  that  we 
shall  pay  attention  to  lunar  parallax  only,  but  this  is  so  important  to 
navigation  that  we  shall  give  it  a  full  explanation. 

The  moon's  horizontal  parallax 
is  given  hi  the  Nautical  Almanac  for 
every  noon  and  midnight  of  Green- 
wich time,  and  from  the  horizontal 
parallax  we  must  deduce  the  paral- 
lax corresponding  to  any  other 
altitude. 

Let  AC  be  the  radius  of  the 
earth,  A  the  position  of  an  observer, 
Z  his  zenith,  and  suppose  H  to  be  the  moon  in  the  horizon ;  then 
the  angle  AHG*  is  the  moon's  horizontal  parallax,  and  the  angle 
AhC  is  the  parallax  corresponding  to  the  apparent  altitude  hAH. 
Draw  Am  parallel  to  Ch,  then  mAH  would  be  the  true  altitude. 

*  From  this  figure  we  draw  the  following  definition  for  horizontal  parallax. 

The  horizontal  parallax  of  any  body  is  the  angle  under  whichthe  semi-diameter  of 
the  earth  would  appear  as  seen  from  that  body.  Of  course  then,  when  the  body  is 
at  a  great  distance  its  horizontal  parallax  must  be  small,  hence  the  sun  and  the 
remote  planets  have  very  little  parallax,  and  the  fixed  stars  none  at  all. 


NAVIGATION.  301 

Let  C/Tand  Ch  be  each  represented  by  -R.  Put^?=  the  hori- 
zontal parallax,  and  x=  the  parallax  in  altitude,  or  the  angle  mAh 
orAkC. 

Now  in  the  triangle  A  CH,  right-angled  at  A,  we  hare 

1  :  sin.j9  :  :  R  :  AC. 
In  the  triangle  A  Ch  we  have 

sin.  CAh  :  sin.  x  :  :  R  :  A  C. 
By  comparing  these  two  proportions,  we  perceive  that 

1  :  sin.  p  :  :  sin.  CAh  :  sin.  x  ' 
Whence,  sin.  x=  sin.  p.  sin.  CAh 

But  sin.  CA h=  cos.hAff,  for  the  sine  of  any  arc  greater  than  90° 
is  equal  to  the  cosine  of  the  excess  over  90°,  hence, 

sin.#=  sin. jo  cos.hAff 

The  lunar  horizontal  parallax  is  rarely  over  a  degree,  commonly 
less,  and  the  sine  of  a  degree  does  not  materially  differ  from  the  arc 
itself,  hence,  the  preceding  equation  becomes  the  following,  without 
any  essential  error. 

That  is,  x—p  cos.  altitude. 

Or,  in  words,  the  parallax  in  altitude  is  equal  to  the  horizontal 
parallax  multiplied  into  the  cosine  of  the  apparent  altitude  (  radius 
being  unity  ). 

EXAMPLES. 

I.  The  apparent  altitude  of  the  moon's  center  after  being  corrected 
for  dip  and  refraction  was  31°  25';  and  its  horizontal  parallax  at 
that  time,  taken  from  a  nautical  almanac,  was  57'  37";  what  was  the 
correction  for  parallax,  and  what  was  the  true  altitude  as  seen  from 
the  center  of  the  earth  ? 

p=5T  37"=3457"  log.      -        -         3.538699 
31°  25' cos.          -         -     9.931152 


s=49'  10"=2950  log.        -         -         3.469851 

Ans.  Cor.  for  parallax  49'  10" 
True  altitude  32°  14'  10" 

2.  The  apparent  altitude  of  the  moon's  center  on  a  certain  occa- 
sion was  42°  17';  and  its  horizontal  parallax  at  the  same  time  was 
58'  12";  what  was  the  parallax  in  altitude,  and  what  was  the  moon's 
true  altitude?  Ans.  Parallax  in  alt.  43'  4" 

True  alt.  43°  0'  4" 


202  CELESTIAL   OBSERVATIONS. 

No  other  examples  of  this  kind  are  necessary,  as  they  will  inciden- 
tally occur  in  several  places  further  on. 

It  now  remains  to  describe  the  instrument  used  for  taking  angles 
at  sea.  We,  therefore,  give  the  following  illustrations  on  the 

QUADRANT     AND     SEXTANT. 

The  quadrant  and  sextant  are  essentially  the  same  instrument, 
and  the  following  is  an  explanation  of  the  principle  on  which  they 
are  constructed. 

Let  ABC  be  a  section  of  a  reflecting  sur- 
face, FB  a  ray  of  light  falling  upon  it,  and 
reflected  again  in  the  direction  BE,  and  BD 
a  perpendicular  at  the  point  of  impact ;  then 
it  is  a  well  known  optical  fact,  that  the  angles 
FBC  and  EBA  are  equal,  and  that  FB,  DBt  and  EB  are  in  the 
same  plane. 

Again,  if  A  0  were  a  reflecting  surface, 
and  a  ray  of  light,  SJ3,  from  any  celestial 
object  S,  were  reflected  to  an  eye  at  E,  the 
image  of  the  object  would  appear  at  8'  on 
the  other  side  of  the  plane,  the  angles  SB  A 
and  ABS',  as  well  as  EBC,  being  equal ; 
and  if  EB  bear  no  sensible  proportion  to 
the  distance  of  S,  the  angles  SES'  and 
SBS'  may  be  considered  as  equal;  for 
their  difference,  BSE,  will  be  of  no  sensible  magnitude. 

Before  we  proceed  to  the  direct  description  of  the  sextant,  it  is 
necessary  to  give  the  following  important 

LEM  MA . 

If  the  exterior  angle  of  a  triangle  be  bisected,  and  also  one  of  the 
interior  opposite  angles,  and  the  bisecting  lines  produced  until  they 
meet,  the  angle  so  formed  witt  be  half  the  other  interior  opposite  angle. 

Let  ABC  be  the  triangle,  and  bisect  the  exterior  angle  A  CD 
by  the  line  CE,  and  the  angle  B  by  the  line  BE. 

The  angle  E  mil  be  half  the  angle  A. 

Let  each  of  the  angles  ACE,  ECD,  be  designated  by  x  (  as  rep- 


THE   PLANE   TABLE.  203 

resented  in  the  figure  ),  and 
each  of  the  equal  parts  of  the 
angle  E  by  y.  Let  A  repre- 
sent the  angle  A,  and  E  the 
angle  E. 

Now  as  the  sum  of  the  three 
angles  of  every  plane  triangle 
is  equal  to  180°;  therefore,  in  the  the  triangle  ABC,  we  have 

-4+2y-f-C=180°  (1) 

Also,  in  the  triangle  EE  0,  we  have 

E+y+C+z=l8Q°  (2) 

Subtracting  (2)  from  (1)  gives  us 

A—  E+  y—  x—Q  (  3  ) 

Whence,  A=E-\-(x—  -y)  (4) 

But  because  x  is  the  exterior  angle  of  the  triangle  ECB 

x—E-{-y         (see  Elementry  Geometry.) 
Or,  (x—y)=E 

This  value  of  (x  —  y)  substituted  in  (4)  gives 

^ 
A=2E,  or  E=-^  Q.  E.  D. 

Another  Demonstration.  —  The  angle  x  being  the  half  of  A  CD  is 
equal  to 


2 

The  angle  x  is  also  equal  to  E-\-y,  because  it  is  the  exterior  angle 
to  the  triangle  EEC. 
Therefore,  by  equality, 


Whence,  E=%A  Q.  E.  D. 


204  CELESTIAL   OBSERVATIONS. 

We  are  now  prepared  to 
show  the  construction  of 
the  sextant  and  quadrant. 

The  instrument  repre- 
sented by  the  annexed  cut 
is  a  quadrant  or  a  sextant, 
according  as  the  arc  con- 
tains 90°  or  120°,  but  each 
actual  degree  of  arc  is 
graduated  to  2°,  and  the 
space  that  covers  90°  is 
really  but  45°,  and  so  on. 

The  reason  why  a  half 
degree  is  counted  and  mark- 
ed as  a  whole  one,  we  are 
about  to  explain. 

AB  C  is  a  firm  plane  sector,  commonly  made  of  metal  or  ebony; 
AJ  is  a  revolving  index  bar,  turning  on  the  center  A,  to  which  is 
attached  a  vernier  scale,  revolving  over  the  graduated  arc. 

The  graduation  commences  at  B.  At  A  is  a  small  plane  mirror, 
perpendicular  to  the  plane  of  the  sector,  it  is  attached  to  the  index  bar 
and  revolves  with  it.  This  is  called  the  index  mirror  or  index  glass. 

At  H  is  another  small  mirror,  half  silvered  and  the  other  half 
transparent.  This  is  called  the  horizon  glass ;  it  might  be  called 
the  image  glass. 

The  horizon  glass  must  be  perpendicular  to  the  instrument,  and 
parallel  to  AB. 

Now  conceive  a  ray  of  light  coming  from  an  object  S,  striking  the 
mirror  A,  the  index  and  mirror  being  turned  so  as  to  throw  the  re- 
flecting ray  into  the  mirror  ff,  this  mirror  agains  reflects  it  toward 
E,  and  an  eye  anywhere  in  the  line  DHvr'ill  see  the  image  of  the 
object  behind  the  mirror  H.  Conceive  the  ray  of  light  from  S  to 
pass  right  through  the  mirror  at  A,  to  meet  the  line  HE\  then,  it 
is  obvious  that  the  angle  SED  measures  the  angle  between  the 
object  £  and  its  image  D. 

Now,  in  the  triangle  AEH,  by  a  little  inspection,  it  will  be  found 
that  HL  bisects  the  exterior  angle,  and  AJ,  the  index,  bisects  one  of 


NAVIGATION.  205 

the  interior  opposite  angles  ;  therefore,  by  the  preceding  lemma,  the 
angle  HLA  is  half  the  angle  at  E,  but  as  AB  and  the  mirror  R  are 
parallel,  the  angle  HLA  is  equal  JAB.  It  is  obvious  that  JAB  is 
measured  by  the  arc  BJ,  or  it  measures  the  angle  at  E,  if  half  de- 
grees on  BC  are  counted  as  whole  ones,  which  was  to  be  shown. 

A  tube,  and  sometimes  a  small  telescope,  is  attached  to  the  bar 
AB,  and  placed  in  the  direction  of  the  line  EH.  This  is  called  the 
line  of  sight. 

THE  ADJUSTMENT  OF  THE  INSTRUMENT. 

When  this  instrument  is  in  adjustment,  the  two  mirrors  are  per- 
pendicular to  the  plane  of  the  sector,  and  are  parallel  to  each  other 
when  0  on  the  vernier  coincides  with  0  on  the  arc.  We  therefore 
inquire :  First 

Is  the  index  mirror  perpendicular  to  the  plane  of  the  instrument  ? 

The  following  experiment  decides  the  question. 

Put  the  index  on  about  the  middle  of  the  arch,  and  look  into  the 
index  mirror,  and  you  will  see  part  of  the  arch  reflected,  and  the 
same  part  direct ;  and  if  the  arch  appears  perfect,  the  mirror  is  in 
adjustment ;  but  if  the  arch  appears  broken,  the  mirror  is  not  in 
adjustment,  and  must  be  put  so  by  a  screw  behind  it,  adapted  to 
this  purpose.  Second, 

Are  the  mirrors  parallel  when  the  index  is  at  0  ? 

Place  the  index  at  0,  and  clamp  it  fast ;  then  look  at  some  well- 
defined,  distant  object,  like  an  even  portion  of  the  distant  horizon, 
and  see  part  of  it  in  the  mirror  of  the  horizon  glass,  and  the  other 
part  through  the  transparent  part  of  the  glass  ;  and,  if  the  whole 
has  a  natural  appearance,  the  same  as  without  the  instrument,  the 
mirrors  are  parallel ;  but,  if  the  object  appears  broken  and  distorted, 
the  mirrors  are  not  parallel,  and  must  be  made  so,  by  means  of  the 
lever  and  screws  attached  to  the  horizon  glass.  Third, 

Is  the  horizon  glass  perpendicular  to  the  plane  of  the  instrument? 

The  former  adjustments  being  made,  place  the  index  at  0,  and 
clamp  it ;  look  at  some  smooth  line  of  the  distant  horizon,  while 
holding  the  instrument  perpendicular ;  a  continued  unbroken  line 
will  be  seen  in  both  parts  of  the  horizon  glass ;  and  if,  on  turning 
the  instrument  from  the  perpendicular,  the  horizontal  line  contimtes 


206  CELESTIAL    OBSERVATIONS. 

unbroken,  the  horizon  glass  is  in  full  adjustment ;  but,  if  a  break  in 
the  line  is  observed,  the  glass  is  not  perpendicular  to  the  plane  of  the 
instrument,  and  must  be  made  so,  by  the  screw  adapted  to  that 
purpose. 

After  an  instrument  has  been  examined  according  to  these  direc- 
tions, it  may  be  considered  as  hi  an  approximate  adjustment  —  a  re- 
examination  will  render  it  more  perfect  —  and,  finally,  we  may  find 
its  index  error  as  follows  : — measure  the  sun's  diameter  both  on  and 
off  the  arch  —  that  is,  both  ways  from  0,  and  if  it  measures  the 
same,  there  is  no  index  error;  but  if  there  is  a  difference,  half  that 
difference  will  be  the  index  error,  additive,  if  the  greater  measure  is 
off  the  arch,  sub  tractive,  if  on  the  arch. 

To  measure  the  altitude  of  the  sun  at  sea. 

Turn  down  the  proper  screen  or  screens,  to  defend  the  eye.  Put 
the  index  at  0,  having  it  loose,  look  directly  at  the  sun  through  the 
tube,  and  you  will  see  its  image  in  the  silvered  part  of  the  horizon 
glass.  Now  move  the  index,  and  the  image  will  drop  ;  drop  it  to 
the  horizon,  and  clamp  the  index. 

Let  the  instrument  slightly  vibrate  each  side  of  the  perpendicular, 
on  the  line  of  sight  as  a  center,  and  the  image  of  the  sun  will  appar- 
ently sweep  along  the  horizon  in  a  circle.  While  thus  sweeping, 
move  the  tangent  screw,*  so  that  the  lower  limb  of  the  sun  will  just 
touch  the  horizon,  without  going  below  it.  The  reading  of  the  index 
will  be  the  altitude  corresponding  to  that  instant,  provided  there  be 
no  index  error. 

To  measure  the  angular  distance  between  two  bodies  as  the  sun  and 
moon,  or  the  moon  and  a  star. 

The  most  brilliant  of  the  two  objects  is  always  reflected  to  the 
other.  Loosen  the  index,  place  it  at  0,  and  direct  the  line  of  sight 
to  the  brighter  object,  and  catch  a  view  of  its  image  in  the  silvered 
part  of  the  horizon  glass. 

Turn  the  plane  of  the  instrument  into  the  plane  between  the  two 
objects  ;  now  move  the  index,  keeping  the  eye  on  the  image,  and 

*  The  screens,  adjusting  screws,  clamp  screw,  and  tangent  screw,  are  not  given 
in  our  description  of  the  instrument,  it  is  not  necessary  to  describe  them;  should 
we  attempt  it,  there  is  danger  that  the  spirit  and  clearness  of  the  description 
would  be  lost  in  the  multitude  of  words. 


NAVIGATION.  207 

bring  it  along  to  the  other  object ;  bring  them  as  near  as  possible, 
then  gently  clamp  the  index. 

Hold  up  the  instrument  again,  in  the  plane  between  the  two  ob- 
jects, and  view  one  object  through  the  transparent  part  of  the 
horizon  glass  ;  and  when  the  instrument  is  in  the  right  position,  the 
image  of  the  other  object  will  appear  also  in  the  same  field  of  view, 
and  then  with  the  tangent  screw,  make  the  limb  of  the  reflected 
object  just  touch  the  other,  as  it  moves  past  it  to  and  fro,  by  the 
gentle  motion  of  the  instrument.  When  the  observer  is  satisfied 
that  he  has  got  the  measure  as  near  as  he  can,  he  cries  out, 
mark,  and  his  assistants  mark  the  time  by  the  watch,  and  the  alti- 
tudes of  the  objects  are  also  marked  for  the  same  time,  if  required, 
and  observers  are  present  to  take  them. 

The  first  experiments  in  the  use  of  this  instrument,  other  than 
measuring  a  simple  altitude,  are  generally  failures,  but  a  little  prac- 
tice will  establish  dexterity,  skill,  and  confidence. 

We  are  now  prepared  to  give  examples  for  finding  latitude. 

Let  it  be  remembered,  that  latitude  is  the  observer's  zenith  dis- 
tance from  the  equator,  and  the  nautical  almanac  gives  the  distance 
of  all  the  heavenly  bodies  from  the  equator,  under  the  name  of 
Declination.  We  can  therefore  observe  our  zenith  distance  from  any 
celestial  object,  and  then  apply  its  declination,  and  we  shah1  have 
our  zenith  distance  from  the  equator,  which  is  the  latitude. 

EXAMPLES. 

1.  On  a  certain  day,  the  meridian*  altitude  of  the  sun's  lower 
limb  was  observed  to  be  31°  44',  bearing  south.  At  that  time  its 
declination  was  7°  25'  8"  south,  semi-diameter  16'  9",  index  error 
-j-2'  12",  height  of  the  eye  17  feet.  What  was  the  latitude  ? 

Ans.  50°  38'  north. 

*  To  obtain  the  meridian  altitude  of  the  sun,  the  observer  commences  obser- 
vations before  noon,  while  the  sun  is  still  rising  ;  driving  the  index  forward  as 
fast  as  the  image  appears  to  rise,  and  there  will  come  a  time,  a  few  minutes  in 
succession,  in  which  the  image  appears  to  rest  on  the  horizon,  neither  rises  nor 
falls,  but  at  length  the  image  will  fall  ;  then  the  observer  knows  that  noon  has 
passed,  and  the  greatest  apparent  altitude  will  be  shown  by  reading  the  index. 


208 


NAVIGATION 


Semi-diameter 
Index  error 
Refraction 
Dip 

Sum 


-f-16'  9"    N.A. 
-j-  2.12 

—  1.31     Table. 

—  4.04    Table, 

-r- 12'46" 


Alt.  ob. 
Correction 
Alt.  Q  's  center 

O's  zenith  dis. 
Dec.  south 

Latitude  north 


31°  44 

12'  46" 

31    56   46 
90° 

58°    3'  14" 

7°  25'    8" 


50°  48'    6" 


In  this  example,  if  the  meridian  altitude  had  been  observed  in  the 
north,  in  place  of  the  south,  what  then  would  have  been  the  ob- 
server's latitude  ?  Ans.  66°  28'  22"  south. 

We  may  note  the  following 

RULE.  —  Subtract  the  corrected  altitude  from  90°.  Then  if  the 
observer  and  the  object  are  both  on  the  same  side  of  the  equator,  add  the 
declination,  but  if  on  different  sides,  subtract  the  declination,  and  the 
sum  or  difference  will  be  the  latitude  of  the  observer. 

Find  the  latitude  from  each  of  the  following  meridian  observations: 


Object. 

Alt.  ob. 

Direc. 

S.D. 

Height. 

Declination. 

Latitude. 

1  Sun  L.  L. 

45027' 

South 

16'  15" 

20  feet 

170  19'  31"  S. 

270    2'51"JV. 

2     «   L.L. 

81043' 

South 

15'  47" 

14 

22°  13'   T'N. 

300  18'  10"  JV. 

3     Jupiter 

730  17' 

South 

17 

24°  10'  13"  S. 

70  22'  52"  S 

4     Saturn 

82°  12' 

North 

17 

12°  9'    &'N. 

40  16'  54"  N. 

5     Sirius 

750   5' 

North 

18 

16°  31'        S. 

31°  30'  26"  S. 

6  Sun  U.  L. 

4QQ42' 

North 

16'  17" 

16 

23022'        -S. 

73°  1'  19"  S. 

1  Sun  L.  L. 

87029' 

South 

16'  17'' 

16 

220  9'        & 

19°  50'  12"  S. 

8  Sun  L.  L, 

15045' 

South 

16'   0" 

16 

4°43'        S. 

69023'16"JV 

In  this  table  L.  L.  indicates  lower  limb,  U.  L.  upper  limb,  S.  D.  semi 
diameter,  N.  north,  £  south,  Direc.  direction.  In  these  examples,  the  instru 
ment  is  supposed  to  have  no  index  error. 

Night  observations  at  sea  are  of  little  value,  for  it  is  very  seldom 
that  the  horizon  can  be  defined,  unless  it  is  in  bright  moon-light,  in 
the  tropical  climates. 

For  this  reason,  very  few  navigators  attempt  to  find  the  latitude, 
by  observations  on  the  planets  and  fixed  stars. 

Occasionally,  however,  when  one  of  the  bright  planets,  or  a  con- 
spicuous fixed  star,  comes  to  the  meridian  in  the  morning  or  evening, 
twilight  observations  can  be  made  on  them,  and  the  latitude  deduced. 

Some  navigators  apply  a  summary  correction  to  the  sun's  lower 
limb,  for  semi-diameter,  dip,  and  refraction,  such  as  is  comprised  in 
the  following  table. 


CELESTIAL    OBSERVATIONS. 


209 


>•? 

Correction  to   be  added  to  the  Observed   Altitude  of  the 

^  c 

II 

Sun's  Lower  Limh,  to  find  the  True  Altitude. 

lo 

5T  CT* 

Height  of  the  Eye  above  the  Sea  in  Feet. 

*   ? 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24   1   -2« 

2s    ,    30       32 

34 

~o~ 
5 

3.8 

3.5 

3.1 

2.8 

2.5 

2.3 

2.1 

1.8 

1.6    14 

1.2 

1.0    0.8   0.6!  0.5 

6 

5.3 

4.9 

4.6 

4.3 

4.0 

3.7 

3.5 

3.3 

3.0 

2.8|  2.6 

2.4    2.2   2.1     1.9 

7 

6.4 

6.0 

5.7 

5.4 

5.1 

4.8 

4.6 

4.4 

4.1 

3.9|  3.7 

3.5    3.3   3.2    3.0 

8 

7.2 

6.8 

65 

6.2 

5.9 

5.7 

5.4 

5.3 

5.0 

4.8|  4.6 

4,4    4.2   4.0 

3.9 

9 

7.9 

7.5 

7.2 

6.9 

6.6 

6.4 

6.1 

5.9 

5.7 

5.5 

5.3 

5.1     4.9;  4.7 

4.5 

10 

8.5 

8.1 

7.8 

7.5 

7.2 

6.9 

6.7 

6.5 

6.2 

6.0 

5.8 

5.6    5.4   5.3 

5.1 

11 

8.9 

8.6 

8.2 

7.9 

7.6 

7.4 

7.2 

6.9 

6.7 

6.5 

6.3 

6.1     5.9i  5.7 

5.6 

12 

9.3 

9.0 

8.7 

8.3 

8.0 

7.8 

7.6 

7.3 

7.1 

6.9 

6.7 

6.5 

6.3 

6.2 

6.0 

14 

9.9 

9.6 

9.2 

8.9 

8.7 

8.4 

8.2 

7.9 

7.7 

7.5 

7.3 

7.1 

6.9 

6.8 

6.6 

16 

10.4 

10.1 

9.7 

9.4 

9.1 

8.9 

8.7 

8.4 

8.2 

8.0 

7.8 

7.0 

7.4 

7.2 

7.1 

18 

10.8 

10.4 

10.1 

9.8 

9.5 

9.3 

9.0 

8.8 

8.6 

8.4 

8.2 

8.0 

7.8 

7.6 

7.5 

20 

11.1 

10.7 

10.4 

10.1 

9.8 

9.6 

9.3 

9.1 

8.9 

8.7 

8.5 

8.2 

8.1 

7.9 

7.7 

22 

11.4 

11.0 

10.7 

10.4 

10.1 

98 

9.6 

9.4 

9.1 

8.9 

8.7 

8,5 

8.3 

8.2 

8.0 

26 

11.7 

11.4 

11.0 

10.7 

10.5 

10.2 

10.0 

9.7 

9.5 

9.3 

9.1 

8.9 

8.7 

8.6 

8.4 

30 

12.0 

11.7 

11.3 

11.0 

10.8 

10.5 

10.3 

10.0 

9.8 

9.6 

9.4 

9.2 

9.0 

8.9 

8.7 

35 

12.3 

11.9 

11.6 

11.3 

11.0 

10.7 

10.6 

10.3 

10.1 

9.9 

9.7 

9.4 

9.2 

9.2 

9.0 

40 

12.5 

12.2 

11.8 

11.5 

11.3 

11.0 

10.8 

10.5 

10.3 

10.1 

9.9 

9.7 

9.5 

9.4 

9.2 

45 

12.7 

12.4 

12.0 

11.7 

11.5 

11.2 

11.0 

10.7 

10.5 

10.210.1 

9.8 

9.7 

9.6 

9.4 

50 

12.8 

12.5 

12.2 

11.9 

ll.fi 

11.3 

11.1 

10.9 

10.6!  io.4:  10.3 

100 

9.8 

9.7 

9.5 

55 

13.0 

12.6 

12.3 

12.0 

11.7 

11.5 

11.2 

11.0 

10.7 

10.5 

10.3 

10.1 

9.9 

9.8 

9.6 

60 

13.1 

12.7 

124 

12.1 

11.8 

11.6 

11.3 

11.1 

10.9 

10.6 

10.4 

10.2 

10.1 

9.9 

9.7 

65 

13.2 

12.8 

12.5 

12.2 

11.9 

11.7 

11.4 

11.2 

11.0 

10.7,105 

10.3 

10.1 

10.0    9.8 

70 

13.3 

12.9 

12.6 

12.3 

120 

11.8 

11.5 

11.3 

11.0 

10.8106 

10.4  10.2 

10.1 

9.9 

75 

13.4 

13.1 

12.7 

12.4 

12.1 

11.9 

11.7 

11.4 

11.2 

11.0 

10.8 

10.6il0.4 

10.2  10.1 

80 

13.6 

13.2 

12.9 

12.612.3 

12.0 

11.8 

11.6 

11.3111.1 

10.9 

10.7 

10.5 

10.41102 

Monthly                Jan. 

Feb. 

Mar. 

April,              May, 

J  une, 

Correction           -fO'  3 

-|-0'.2 

-fO'.l 

O'O           —O'.l 

—  0'.2 

for  Sun's               July. 

Aug. 

Sept. 

Oct.                 Nov. 

Dec. 

Semi-diam.            —  0'.3 

—  0'.2 

—  O'.l 

-J-0'.l          -t-0'.2 

-t-0'.3 

The  most  practical  method  of  obtaining  the  latitude  by  observa- 
tion, other  than  the  meridian  altitude  of  the  sun,  is  by  the  meridian 
altitude  of  the  moon ;  but  to  correct  the  observed  altitude  for 
semi-diameter,  parallax,  refraction,  and  dip,  and  do  it  to  the  utmost 
accuracy,  requires  more  computation  and  attention  than  the  mere 
practical  navigator  is  disposed  to  give.  Moreover,  such  like  accuracy 
is  not  required  in  practical  navigation.  To  know  the  latitude  within  a 
mile  is  all  the  ship  master  requires  ;  and  this  can  be  done  in  a  very 
summary  manner,  by  observing  the  moon's  meridian  alititude  and 
using,  the  following  tables,  according  as  the  lower*  or  upper  limb 
of  the  moon  is  observed. 


*  The  bright  limb,  is  the  one  observed,  whether  it  be  the  upper  or  lower. 
14 


210 


NAVIGATION. 


These  tables  make  but  one  correction  for  semi-diameter,  parallax  and  refraction 


TABLE  I. 

CORRECTIONS  to  be  added  to  the  OBSERVED  ALTITUDE  of  the  Moon'*  lower  limb. 

Fart  let  HORIZONTAL  PARALLAX. 

d'B 
Alt 

53' 

54' 

by 

66' 

67' 

68' 

w 

w 

61' 

6 

0.59 

.  0 

1.  1 

.  3 

.  4 

1.  5 

.  6 

1.  8 

1.  9 

8 

1.  0 

.  2 

1.  3 

.  4 

.  6 

1.  7 

.  8 

1.  9 

1.11 

10 

1.  1 

.  3 

1.  4 

.  5 

.  7 

1.  8 

.  9 

1.10 

.12 

15 

1.  2 

.  3 

1.  5 

.  6 

.  7 

1.  9 

.10 

1.11 

.12 

20 

1.  2 

.  3 

1.  4 

.  5 

.  6 

1.  8 

.  9 

1.10 

.11 

25 

1.  0 

.  2 

1.  3 

.  4 

.  5 

1.  6 

.  7 

1.  9 

.10 

30 

0.59 

1.  0 

1.  1 

.  2 

.  3 

1.  4 

.  5 

1.  7 

.  8 

35 

0.57 

0.59 

0.59 

.  0 

1.  1 

1.  2 

.  3 

1.  4 

.  5 

40 

0.55 

0.55 

0.56 

0.57 

0.58 

0.59 

.  0 

1.  1 

1.  2 

45 

0.51 

0.52 

0.53 

0.54 

0.55 

0.56 

0.57 

0.58 

0.59 

50 

0.48 

0.49 

0.50 

0.51 

0.51 

0.52 

0.53 

0.54 

0.55 

55 

0.44 

0.45 

0.46 

0.47 

0.48 

0.49 

0.49 

0.50 

0.51 

60 

0.40 

0.41 

0.42 

0.43 

0.44 

0.44 

0.45 

0.46 

0.47 

65 

0.36 

0.37 

0.38 

0.39 

0.39 

0.40 

0.40 

0.41 

0.42 

70 

0.33 

0.33 

0.34 

0.34 

0.35 

0.36 

0.36 

0.37 

0.37 

75 

0.28 

0.28 

0.29 

0.29 

0.30 

0.30 

0.31 

0.31 

0.32 

80 

0.24 

0.24 

0.24 

0.25 

0.25 

0.25 

026 

0.26 

0.27 

85 

0.19 

0.19 

0.19 

0.20 

0.20 

0.21 

0.21 

0.21 

0.22 

TABLE  II.   CORRECTIONS  to  be  applied  to  OBSERVED  ALTITUDE  of  the  Moon's  upper  limb 

Part  2nd.  HORIZONTAL  PARALLAX. 

j;« 

63' 

64' 

55' 

56' 

57' 

68' 

59' 

eo' 

61' 

Alt. 

o    / 

o    / 

0      / 

O      / 

o    / 

o    / 

o    / 

o    / 

0      1 

To 

15 

4-0.33 
0.33 

4-0.33 
0.33 

4-0.34 
0.35 

4-0.34 
0.35 

+0.34 
0.36 

-}-0.36 
0.37 

-j-0.37 
0.37 

+  0.37 
0.39 

+0.38 
0.39 

20 

5.32 

0.33 

0.34 

0.35 

0.35 

0.36 

0.37 

0.37 

0.38 

26 

0.30 

0.32 

0.32 

0.33 

0.33 

0.34 

035 

0.35 

0.36 

30 

0.29 

0.30 

0.31 

0.31 

0.32 

0.32 

0.33 

0.34 

0.34 

36 

0.26 

0.26 

0.27 

0.27 

0.28 

0.28 

0.29 

0.29 

0.30 

40 

0.24 

0.25 

0.26 

0.26 

0.27 

0.27 

0.28 

0.29 

0.29 

46 

0.19 

0.22 

0.22 

0.22 

0.23 

0.24 

0.24 

0.24 

0.26 

50 

0.17 

0.19 

0.19 

0.20 

0.20 

0.21 

0.21 

0.21 

021 

55 

0.14 

0.15 

0.16 

0.16 

0.16 

0.16 

0.17 

0.17 

017 

60 

0.10 

0.11 

0.12 

0.12 

0.12 

0.12 

0.13 

0.13 

0.13 

65 

0.  6 

0.  7 

0.  7 

0.  8 

0.  8 

0.  8 

0.  8 

0.  8 

0.  9 

70 

0.  3 

0.  3 

0.  3 

0.  3 

0.  3 

0.  3 

0,  3 

0.  3 

0.  4 

75 

—0.  1 

—  0.  1 

—  0.  1 

—0.  1 

—  0.  1 

-0.  2 

—  0.  2 

—  0.  2 

-0.  2 

80 

0.  6 

0.  6 

0.  6 

0.  6 

0.  6 

0.  6 

-0.  6 

0.  7 

0.  7 

85 

0.10 

0.11 

0.11 

0.11 

0.11 

0.11 

0.11 

0.12 

0.12 

Height  of  the  eye,                                            4ft.         9ft         16ft.    |     26ft        36ft. 

Dip  of  the  Horizon,                                         —2'  )    —3     |    —4'    }    —b'    \    —6' 

EXAMPLES. 


1.  In  longitude  about  45°' west,  on  the  5th  of  January,  1852,  at 
about  llh.  in  the  evening,  I  observed  the  altitude  of  the  moon's 
lower  limb  as  she  passed  the  meridian,  and  found  it  to  be  68°  12* 
from  the  south,  height  of  the  eye  16  feet.  What  was  my  latitude  ? 


CELESTIAL    OBSERVATIONS.  211 

On  the  5th  bf  Jan.,  at  llh.  evening,  Ion.  45  west,  corresponds  to  2  hours 
after  midnight  at  Greenwich. 

From  the  Nautical  Almanac,  we  find,  that, 

At  midnight  of  the  5th,  the  moon's  horizontal  parallax  was  -        -  57'  4</' 

At  noon  of  the  6th,        •        * 58*    1" 

Therefore,  by  proportion,  the  horizontal  parallax  at  the  time  of  observation, 
must  have  been  57' 43". 

Moon's  declination  at  midnight  of  the  5th,  (N.  Almanac),       21°  47'  53"  N. 
"  "  noon  of  the  6th,    -       -       -        -       22°  16'  55"  N 

Variation  in  12  hours, 29'   2" 

Therefore,  the  variation  for  2  hours,  was  not  far  from    •  4'  50" 

Hence  the  dec.  at  the  time  of  observation  was,    -  22°  52'  43"  N. 

We  enter  table  1,  and  under  the  parallax,  and  opposite  to  the 
altitude  as  near  as  we  can  find  them,  we  perceive  that  37'  must  be 
about  the  correction  for  the  altitude. 

Whence,         Observed  alt.  L.  L 
Correction, 

Dip.  always  sub.         — 


Zenith  dis. 
3)  'a  dec. 


Lat.  in  44°    8'     North. 

Find  the  true  altitude  of  the  moon's  center,  in  each  of  the  follow- 
ing examples.     L.  L.  means  lower  limb  ;   U.  L.  upper  limb. 


Height  of 

Ans. 

Obserred  Alt. 

H.  P. 

the  eye. 

True  Alt. 

1. 

3  L.L. 

53°  23' 

58'  14" 

14  feet 

54°  10'  nearly. 

2. 

D  L.L. 

.    48°  58 

60'  27" 

19    " 

49°  48'      " 

3. 

D  U.L. 

57°  11' 

54'  30" 

20    " 

57°  19'      " 

4. 

J  L.L. 

63°  38' 

65'  29" 

12    " 

64°  14'      " 

5. 

J  U.L. 

20°    3' 

54'  14" 

16    " 

20°  32'      " 

6. 

T>  L.L. 

16°    2' 

59'  38" 

23    " 

17°  12'      " 

When  the  weather  makes  it  doubtful  whether  meridian  observa- 
tions can  be  obtained,  navigator's  resort  to  double  altitudes,  or  to  the 
altitudes  of  two  objects  taken  at  the  same  time.  We  shall  only  show 


212  NAVIGATION. 

the  principle  on  which  this  method  is  founded  ;  it  is  the  application 
of  spherical  trigonometry. 

Let  Pic  be  the  earth's  axis,  Qq 
the  equator.  Suppose  the  sun  to  be 
the  object,  and  let  its  position  be  S 
and  T  at  two  different  times. 

The  elapsed  time  measures  the 
angle  SPT.  In  the  triangle  PTS, 
we  have  the  two  sides  PT,  PS,  and 
the  included  angle,  from  which  we 
compute  the  side  TS,  and  the  angle 
TSP. 

Subtracting  the  altitudes  Sm  and  Tn  from  90°,  we  have  ZS,  and 
ZT,  then  we  have  all  the  sides  of  the  triangle  ZTS,  from  which  we 
compute  the  angle  TSZ.  Subtracting  this  angle  from  TSP,  gives 
us  the  angle  ZSP.  Now,  in  the  triangle  ZSP,  we  have  the  two 
sides  ZS,  SP,  and  their  included  angle,  from  which  we  compute  PZ 
the  complement  of  the  latitude. 

If  the  ship  sails,  during  the  interval  between  the  observations,  a 
correction  will  be  required  for  the  first  altitude,  and  such  corrections 
are  found  by  the  traverse  table  ;  a  nautical  mile  in  the  direction  of 
the  sun,  corresponds  to  one  minute  of  a  degree,  to  be  applied  to  the 
altitude.  When  the  proper  correction  is  made,  the  result  is  equiva- 
lent to  having  both  altitudes  taken  at  the  last  station,  and  the  deduced 
latitude  is  the  latitude  of  that  station. 


CHAPTER    II. 

LONGITUDE. 

LONGITUDE,  from  celestial  observations,  is  measured  by  time.  A 
place  15°  west  of  another,  will  have  noon  one  hour  of  absolute 
tune  later ;  if  30°  west,  the  local  time,  noon  will  be  two  hours  later, 
<kc.,  <fec.;  15°  corresponding  to  an  hour  in  time.  Therefore,  if  we 
have  any  way  of  determining  the  times  at  two  places,  correspond- 
ing to  the  same  absolute  instant,  the  difference  of  such  times  will 


LONGITUDE.  213 

correspond  to  the  difference  of  longitude  between  the  two  places  at 
the  rate  of  15°  to  an  hour,  or  4  minutes  to  a  degree. 

A  perfect  time  piece  will  keep  the  time  at  any  particular  meridian, 
and  by  carrying  that  perfect  time  piece  with  us,  by  it  we  can  see  the 
time  at  that  particular  meridian  ;  and  then  if  we  can  find  the  time  at 
the  place  where  we  are,  the  comparison  of  these  two  times  will  give  the 
difference  of  longitude,  that  is,  the  difference  between  our  longitude 
and  that  of  the. particular  meridian,  to  which  the  time  piece  refers. 

For  instance,  a  gentleman  leaves  Boston  ;  his  watch  is  a  perfect 
time  piece,  and  it  is  set  to  Boston  time,  he  travels  west  on  the  rail- 
roads, his  watch  all  the  while  shows  Boston  time  ;  when  it  is  twelve 
o'clock  by  his  watch  it  is  really  so  in  Boston,  but  not  so  at  the  place 
where  he  is.  The  sun  has  arrived  at  the  meridian  of  Boston, 
but  not  yet  at  the  meridian  of  Albany,  or  Buffalo,  or  Detroit ;  and 
when  the  gentleman  arrives  at  any  of  these  places,  or  any  interme- 
diate place,  the  local  time,  compared  with  the  time  hi  Boston,  will 
give  the  longitude  of  that  locality  from  Boston,  counting  one  degree 
for  every  four  minutes  in  the  difference  of  time. 

Unfortunately,  however,  there  is  no  such  thing  as  a  perfect  time 
piece,  but  some  do  approximate  toward  perfection.  Such  ones,  made 
with  the  greatest  care  and  solely  for  accuracy  in  rate  of  motion,  are 
called  chronometers  ;  th£y  are  supposed  to  keep  time  within  certain 
known  limits,  and  in  the  place  of  perfect  time  keepers,  they  are  used 
at  sea  for  finding  longitude. 

Chronometers  show  the  time  at  the  distant  place,  it  then  remains 
to  find  the  time  at  ship,  and  this  is  done  most  accurately  by  spheri- 
cal trigonometry,  as  will  soon  appear. 

The  sun's  altitude  is  greatest  just  at  apparent  noon,  but  by  obser- 
vations we  cannot  define  just  the  moment  when  that  takes  place  ; 
hence  meridian  observations,  valuable  as  they  are  for  latitudes,  are 
worth  nothing  for  time,  when  time  is  to  be  settled  to  anything  like 
accuracy. 

The  best  position  of  the  sun  ( or  any  other  celestial  object ) 
for  an  observation  to  find  local  time,  is  when  it  is  nearly  east  or  west, 
and  its  altitude  more  than  ten  degrees. 

In   such    circumstances,  an   observer    can   find  the  local  time 


214  NAVIGATION. 

within  5  or  6  seconds,  by  taking  an  altitude  of  the  sun,  provided  he 
at  the  same  time  knows  his  latitude  and  the  sun's  polar  distance. 

The  operation  is  a  beautiful  application  of  spherical  trignometry, 
and  it  is  illustrated  by  the  following  figure. 

Let  Z  be  the  zenith  of  the 
observer,  P  the  pole,  S  the 
position  of  the  sun,  and  PS 
the  sun's  polar  distance.* 

When  S  comes  on  to  the 
meridian,  it  is  then  apparent 
noon ;  and  the  angle  ZPS  of 
the  triangle  ZPS  measures 
the  interval  from  apparent 
noon,  at  the  rate  of  four  min- 
utes to  one  degree. 

The  side  PS  is  the  polar 
distance,  the  side  ZS  is  the  co-altitude,  and  the  side  PZ  is  the  co- 
latitude. 

Now,  in  every  treatise  on  spherical  trigonometry,  it  is  demon- 
strated as  a  fundamental  principle,  that 

The  cosine  of  any  angle,  of  a  spherical  triangle,  is  equal  to  the  co- 
sine of  its  opposite  side,  diminished  by  the  rectangle  of  the  cosines  of  the 
adjacent  sides,  divided  by  the  rectangle  of  the  sines  of  the  adjacent 
sides. 

cos.  ZS—  cos.  PZ  cos.  PS 

That  IS,       COS.    P=    : TTj->-         D  ry 

sin.  PZ  sin.  PS 

Now,  in  place  of  cos.  ZS,  we  take  its  equal,  sin.  ST,  or  the  sine 
of  the  altitude,  and  in  place  of  cos.  PZ,  we  take  its  equal,  the  sine 
of  the  latitude. 

In  short,  let  4=  the  altitude,  Z=  the  latitude,  and  D=  the 
polar  distance. 

sin.  A — sin.  L  cos.  D 


Then         cos.  P== 


cos.  L  sin.  D 


*  When  the  observer  is  in  the  northern  hemisphere,  the  polar  distance  is 
counted  from  the  north  pole-;  when  in  the  southern  hemisphere,  from  the  south 
oole. 


LONGITUDE.  215 

From  a  general  equation,  in  plane  trigonometry,  we  have 

2sin.2  £P=1— cos.  P 
Substituting  the  value  of  cos.  P,  in  this  last  equation,  we  have 

sin.  A — sin.  L  cos.  D 

2  sin.2  4.  P=l— Y-. „ — 

cos.  L  sm.  D 

^(cos.  L  sin.  J9-fsin.  L  cos  D) — sin.  A 
cos.  L  sin.  D 

By  comparing  the  quantity  in  parentheses  with  eq.  (7),  plane 
trigonometry,  we  perceive  that 

sin.  (L+D)—  sin.  A 

2  sin.2  A  P=  * j2-. 7, — 

cos.  Lsm.  D 

Considering  (L-{-D)  to  be  a  single  arc,  and  then  applying 
equation  (16),  plane  trigonometry,  and  dividing  by  2,  we  shall 
have 


/L+D+A\         (L+D—A 
cos.  ^     — 2 J  sin.  ^ g 

cos.  Z  sin  D. 


sin2,     P= 


But  L+D-A  =  L+!>+A  _A,  and  now  if  we  put 


S=  -  g  -    we  shall  have 

cos.  Ssin.  (S—A) 
™*'lp=       cos./,  sin.  D 


Or,  sin.  i  P=        cos.    -  sn.       - 


cos.  L  sm.  X> 

This  is  the  final  result  when  radius  is  unity,  when  it  is  R  times 
greater,  thjp  the  sin.  -£•  P  will  be  R  times  greater,  and  if  R  repre- 
sents the  radius  of  our  tables,  to  correspond  with  these  tables  we 
must  multiply  the  second  member  by  R,  and  if  we  put  it  under  the 
radical  sign,  we  must  multiply  by  R2  ;  in  short  we  shall  have, 


Sin.  |  P=J(J*\  (--?-=\  cos.  8  sin.  (&-A) 
^   \cos.  LJ  \sm.  D/ 


The  right  hand  member  of  this  equation,  shows  four  distinct 

7") 

garithms ;  thus,  is  the  cosine  of 

cos.  L 
10,  which  we  shall  call  cosine  complement. 


n 

logarithms ;  thus,  is  the  cosine  of  the  latitude  subtracted  from 

cos.  L 


216  NAVIGATION. 

This  equation  furnishes  the  following  rule  for  finding  apparent 
local  time,  when  the  sun's  altitude,  its  polar  distance,  and  the  lati- 
tude of  the  observer,  are  given. 

The  altitude  must  be  observed,  the  latitude  must  be  known,  and  the 
Nautical  Almanac  will  furnish  the  polar  distance. 

RULE. — 1.  Add  (off ether  the  altitude,  latitude,  and  polar  distance;  take 
the  half  sum,  and  from  the  said  half  sum  subtract  the  altitude,  thus 
finding  the  remainder. 

2.  The  logarithms.     Find  the  cosine  complement  of  the  latitude,  the 
sine  complement  of  the  polar  distance,  the  cosine  of  the  half  sum,  and 
the  sine  of  the  remainder. 

3.  Add  these  four  logarithms  together  >  and  divide  by  2,  the  logarithm 
thus  found,  is  the  sine  of  half  the  polar  angle,  or  half  the  sun's 
meridian  distance. 

4.  Take  out  the  arc  corresponding  to  this  sine,  and  divide  its  doiible 
by  15  (as  in  compound  division  in  arithmetic),  and  the  quotient  mil 
be  the  hours,  minutes,  and  seconds  from  apparent  noon  ;  and  if  the  sun 
is  east  of  the  meridian,  the  hours,  minutes,  and  seconds,  must  be  sub- 
tracted from  1 2  hours,  for  the  corresponding  time  of  day. 

The  time  shown  by  a  chronometer  or  a  perfect  clock,  or  rather 
graduation  of  clocks,  is  to  mean  and  not  to  apparent  time,  and  to 
convert  apparent  into  mean  time,  the  equation*  of  time  is  given  in 
the  Nautical  Almanac  for  the  noon  of  every  day  at  Greenwich.  The 
amount  of  it,  reduced  or  modified  to  correspond  to  the  time  of  obser- 
vation, can  be  applied  to  apparent  time,  and  the  mean  tune  of  taking 
the  observation  will  be  determined.  The  difference  between  this 
time  and  the  mean  time  at  Greenwich,  as  determined  by  the  chro- 
nometer, will  be  the  longitude.  The  longitude  will  be  west,  if  the 
time  at  Greenwich  is  latest  in  the  day,  otherwise  it  will  be  east. 

If  the  observer  is  on  land,  without  a  sea  horizon,  and  uses  a 
reflecting  instrument,  he  must  have  an  artificial  horizon.  A  proper 
artificial  horizon,  is  a  small  dish  of  mercury,  with  a  glass  roof  to  put 
over  it,  to  keep  the  mercury  from  being  agitated  by  the  wind.  In 
place  of  the  mercury,  a  plate  of  molasses  will  answer.  In  still  calm 
weather  any  clear  pool  of  water  is  a  good  artificial  horizon. 

In  either  of  these,  the  reflected  image  of  the  object  appears  as 
much  below  the  horizon  as  it  is  above  it,  and  to  measure  the  altitude, 

*  For  the  theory  of  equation  of  time,  see  works  on  astronomy. 


LONGITUDE.  217 

the  image  reflected  by  the  mirror  of  the  instrument  must  be  carried 
to  the  image  in  the  artificial  horizon  ;  half  of  the  angle  shown  by 
the  index  will  be  the  apparent  altitude.  In  using  an  artificial  hori- 
zon there  is  no  dip,  other  corrections  are  to  be  applied  according  to 
circumstances.  • 

EXAMPLES     UNDER     THE     PRECEDING   RULE. 

1.  Being  at  sea,  May  20th,  1823,  in  latitude  43°  30'  N.,  and  in 
longitude  about  20°  west,  I  observed  the  altitude  of  the  sun's 
lower  limb,  and  found  it  to  be  32°  4'  rising,  when  an  assistant 
marked  the  time  per  watch,  at  7h.  43m.  A.  M. ;  height  of  the  eye 
16  feet.  What  was  the  true  mean  time  ? 

Just  before  the  observation,  the  watch  was  compared  with  the 
chronometer  in  the  cabin*,  and  found  to  be  1  hour,  21  minutes,  and 
12  seconds  slow  of  chronometer. 

On  the  8th  of  May,  the  chronometer  was  3m.  7s.  fast  of  Greenwich 
time,  and  gaining  ls.6  daily.  What  was  the  longitude  ? 


QS.D.,      - 

4-   15'  49" 
3  56 

H.  M.  8- 

Watch,  -  -  -  -  743  0 

Ref.,   - 

Correction, 
Observation, 

Alt.  O  center, 

—     1  30 

-    -f   10  23 
32     4 

Face  of  chron.  at  ob.,  -  9  4  12 
Error  3m.  7s.,  increase  of  error 

-    32°  14'  23" 

Greenwich  time,  -  -  9  0  46 

At  noon  on  the  20th  of  May,  1823,  the  sun's  declination,  by  the  N.  A.,  was 
19°  52'  18"  north,  increasing  at  the  rate  of  30".  6  per  hour,  and   the  time  of 
taking  the  observation  was  3  hours  before  noon  at  Greenwich  ;  therefore,  the 
declination  must  have  been  19°  50'  47"  N. 
Altitude,        32°  u'  23" 


Lat., 
P.P., 

S. 

(S-A) 

43  30 
70  9  13 

cos.  com. 
sin.  com. 

cosine 
sine 

.139435 
.026603 

9.467253 
9.814363 

2)145  53  36 

72  56  48 
32  14  23 

40  42  25 

2)19.447657 

31°  58'  8" 

sin. 

9.723828 

»  Chronometers  should  never  be,  and  by  careful  persons,  never  are,  taken  oat 
of  their  places  during  a  voyage. 


218  NAVIGATION. 

31°  58' 8» 


63°  56' 16"=  4h.  15m.  45s. 
12 


Apparent  time,        -        -        -  7  44     15    A.  M. 

Equation  of  time  N.  A.       -  —       3     51 

Mean  time  at  ship,          -        -  7  40      24 

Watch,      .....  7  43 
Watch  too  fast,      ...                2m.  36s7~ 

Time  at  Greenwich  per  ohron.,  9A.    Om.  46s. 

Time  at  ship  per  observation,  7  40     24 

Diff., 1    20      22=2(P  5' west  Ion. 

2.  August  10th,  1824,  in  latitude  54°  12'  north,  at  5h  33m  per 
watch,  height  of  the  eye  18  feet,  I  observed  the  altitude  of  the 
sun's  upper  limb  16°  50'  falling.  My  chronometer  was  2h  20m 
37s  fast  of  the  watch  ;  and  on  the  7th,  the  same  month,  the  chro- 
nometer was  40m  29.4s  fast  of  Greenwich  time,  gaining  7/^  seconds 
daily.  What  was  the  error  of  the  watch,  and  the  longitude  per 
chronometer  ? 

Preparation. 

Time  per  watch          -        5A.  33m.  Os.    P.  M. 
Diff.  per  chron.      -        -    2    20    37 

Face  of  chronometer   -       7    53    37      P.  M. 
Chron.  fast  (whole  error)       — 40    52 

Greenwich,  mean  time,      7     12    45      P.  M. 

On  the  10  of  August,  1824,  the  sun's  declination  at  noon,  Greenwich  time, 
was  15°  32'  14"  north,  decreasing  at  the  rate  of  45"  per  hour,  as  given  in  the 
Nautical  Almanac.  The  decrease  for  7|  hours  must  be  5'  24";  whence,  the 
declination  at  the  time  of  observation,  15°  26'  50"  N.,  and  the  polar  distance 
74°  33'  10". 


Observed  altitude        -  16O  50'  00" 

Semi-diameter,  N.  A.      -  —15  48 

DipandRef.        -        -  —1  20 

True  alt.  center       -        -  16O  26'  52" 


Equation  of  time,  per  N.  A.,  Aug. 
10,  1824,  was  -  -  -f  5m.  2s. 
Hourly  decrease  y^s.  — 2 

Equation  at  ob.         -         -        5m.  Os. 


We  now  leave  the  problem  to  be  worked  through  by  the  pupil,  giving  only 
the  answer. 

Ans.  Watch  slow  of  local  mean  time,  3m  27s. 
Longitude  by  chronometer,  24°  4'  30"  west. 
3.  When  it  was  6h  Om  21s,  P.  M.,  mean  time,  at  Greenwich,  by 
my  chronometer,  I  observed  the  altitude  of  the  sun's  lower  limb  to 
be  30°  17',  in  the  afternoon  of  January  12th,  1862.     At  noon  our 


LONGITUDE.  219 

latitude,  by  a  meridian  observation,  was  21°  47'  north,  and  since 
that  time  we  have  made  11  miles  of  southing,  by  the  log.  The 
dip  was  4',  and  semi-diameter  16'  17".  What  was  the  longitude  by- 
chronometer  ? 

Sun's  Declination  Jan.  12,  '52,  at  noon,  G.  T.    -    21°  44'  10"  south. 
Hourly  decrease,  per  N.  A.,  25",  giving        -  — %  30" 

Declination  at  the  time  of  observation      -         -    21°  41" 40"  south 
Equation  of  time  at  noon,  Greenwich      -        -         -|-8m.  25s. 
Hourly  increase  T9ff<^  of  a  second,  making    -  6*.  nearly 

Equation  at  time  of  observation  (to  add)          -  -f-8m.  31s. 

Were  we  sure  that  pupils  would  have  access  to  nautical  almanacs,  we  would 
give  neither  declination  nor  equation  of  time. 

Ans.  Lon.  46°  9'  west. 

4.  On  the  16th  of  January,  1852,  when  my  chronometer  showed 
llh  27m  41s,  A.  M.,  for  the  mean  time  at  Greenwich,  I  observed  the 
altitude  of  the  sun's  lower  limb  and  found  it  32°  21'  rising,  height 
of  the  eye  16  feet,  latitude  0°  41'  south.  What  was  the  longitude 
by  chronometer  ? 

By  the  N.  A.,  the  sun's  declination  at  that  time  was  21°  2  36''  south,  and 
the  equation  of  time  9m.  53s.  additive. 

Ans.  Lon.  44°  39'  west. 

N.  B.  —  Time  at  any  place,  is  but  the  difference  between  the 
right  ascension  of  the  meridian  and  the  right  ascension  of  the  sun  ; 
and  to  find  the  time  from  these  two  elements,  we  always  subtract 
the  right  ascension  of  the  sun  from  the  right  ascension  of  the  meri- 
dian, increasing  the  latter  by  24  hours,  to  render  subtraction  possi- 
ble, when  necessary. 

The  right  ascensions  of  the  stars  are  given,  and  the  right  ascen- 
sion of  the  sun  is  given,  in  the  Nautical  Almanac,  for  the  noon  of 
every  day  in  the  year,  Greenwich  time.  Now,  if  we  can  find  the 
meridian  distance  of  any  known  star,  by  observation,  we  can  estab- 
lish the  right  ascension  of  the  meridian,  and,  consequently,  the  local 
time.  Hence,  we  can  find  longitude  by  comparing  the  chronometer 
with  the  altitudes  of  the  stars,  aa  the  following  example  will  illus- 
trate. 

6.  If  on  the  8th  of  March,  1 862.  when  my  chronometer  showed  the 
Greenwich  time  to  be  7h  22na  3a,  P.  M.,  I  found  by  observation, 


220  NAVIGATION 

that  the  true  altitude  of  Sirius  was  37°  52'  west  of  the  meridian. 
My  latitude  was  32°  28'  south.  What  was  the  time  at  ship,  and 
what  was  my  longitude  ;  the  elements  for  computation  being  as 
follows  ? 

1.  Right  ascension  of  the  star      -        -        6h.  38m.  38s. 

2.  Declination  of  the  star  16°  31'  south 

3.  Right  ascension  of  the  sun     -         -         23A.  1 1m.  25s. 
By  means  of  the  triangle  we  find, 

The  meridian  distance  of  the  star        -         3h.  40m.  58s. 
To  which  add  -fc's  R.  A.,  because  -fc  is  west  6     38      38 

Right  ascension  of  the  meridian      -        -   10     19      36 
Add 24 


34     19      36 
Subtract  the  R.  A.  of  the  sun        -        -     23     17      25 


Diff.  is  apparent  time  at  ship      -  -         11  2  1 1  P.  M. 

Equation  of  time,  add    -               -  -  10  48 

Mean  time  at  ship     -        -        -  -         11  12  59 

Tune  at  Greenwich        ...  -     7  22        3 


Longitude  in  time       -        -        -        -        3     50      56 

=57°  44'  east 

N.  B. —  When  the  chronometer  remains  in  the  same  place  for  a 
week  or  more,  its  rate  can  be  determined  by  comparing  it  with  the 
observed  altitudes  of  the  sun,  taken  from  day  to  day.  In  different 
climates  the  same  chronometer  will  have  different  rates,  and  on  re- 
turning to  its  original  station  it  will  frequently  resume  its  original 
rate. 

For  azimuths,  and  variations  of  the  compass,  see  page  106. 


CHAPTER    III. 

LUNAR    OBSERVATIONS. 

A  GOOD  and  well-tried  chronometer  is  a  valuable  and  reliable 
instrument  for  finding  the  longitude  at  sea,  during  short  runs  ;  but 
still  it  is  but  an  instrument,  and  is  not  one  of  the  reliable  works  of 


LUNAR    OBSERVATIONS.  221 

nature.  Near  the  end  of  a  long  voyage,  the  best  of  chronometers 
very  frequently  give  false  longitude,  and  in  such  cases,  good  navi- 
gators always  resort  to  lunar  observations,  which  from  the  hands  of 
a  good  observer,  can  be  relied  upon  to  within  10  or  12  minutes  of  a 
degree,  and  they  usually  come  within  5  or  6  miles,  and  sometimes 
even  more  exact,  but  that  is  accidental  and  unfrequent. 

To  comprehend  the  theory  of  lunars,  we  must  call  to  mind  the  fact 
that  the  moon  moves  through  the  heavens,  apparently  among  the 
stars,  at  the  rate  of  more  than  13°  in  a  day,  and  any  angular  dis- 
tance it  may  have  from  the  sun  or  any  star  corresponds  to  some 
moment  of  Greenwich  time. 

About  three  days  before  and  after  the  change  of  the  moon,  she  is 
too  near  the  sun  to  be  visible,  but  at  all  other  times,  her  distance 
from  the  sun,  some  of  the  larger  planets,  and  certain  bright  fixed 
stars,  called  lunar  stars,*  which  lie  near  her  path,  are  computed  and 
put  down  in  the  nautical  almanac,  for  every  third  hour  of  mean 
Greenwich  time  commencing  at  noon.  For  any  particular  day,  the 
distances  are  given  to  such  objects  only,  east  and  west  of  her,  as 
will  be  convenient  to  measure  with  the  common  instruments. 

^  —      i    '    ,     \*       •}  •  i  •-       ;*f-.      *•' 

The  distances  put  down  in  the  nautical  almanac,  are  such  as  would 
be  seen  if  viewed  from  the  center  of  the  earth  ;  but  observers  are 
always  on  the  surface  of  the  earth,  and  the  distances  thence  observed, 
must  always  be  reduced  to  equivalent  distances  seen  from  the  center, 
and  this  reduction  is  called  working  a  lunar,  which  is  generally  the 
highest  scientific  ambition  of  the  young  navigator.  \ 

The  true  distance  between  the  sun  and  moon,  or  between  a  star 
and  the  moon,  can  be  deduced  from  the  apparent  distance  by  the 
application  of  spherical  trigonometry. 

The  moon  is  never  seen  by  an  observer  in  its  true  place,  unless  the 
observer  is  in  a  line  between  the  center  of  the  earth  and  the  moon, 
that  is,  unless  the  moon  is  in  the  zenith  of  the  observer ;  in  all  other 

*  There  are  nine  lunar  stars,  Arietis,  Aldebaran,  Pollax,  Regulus,  Spica, 
Antares,  Aquilae,  Fomalhaut,  and  Pegasi. 

. .:    .  \,.*f\ ;.•-''•-?'-     !'  ;".'.  ";'-V  *.:  ;    J 

t  Many  navigators,  both  old  and  young,  direct  all  their  efforts  to  knowing  how 
to  do,  without  attempting  to  comprehend  the  reasons  for  so  doing  ;  and  this  the 
world  calls  practical,  —  a  complete  perversion  of  the  term.  On  the  other 
hand,  some  men  of  the  schools  spend  their  energies  in  metaphysical  nothings, 
splitting  hairs  in  logic,  and  calling  it  scientific  ;  this  is  equally  a  perversion* 


222  NAVIGATION. 

positions,  the  moon  is  depressed  by  parallax,  and  appears  nearer  to 
those  stars  that  are  below  her,  and  further  from  those  stars  that  are 
above  her,  than  would  appear  from  the  center  of  the  earth.  There- 
fore, the  apparent  altitudes  of  the  two  objects,  must  be  taken  at  the 
same  time  that  their  distance  asunder  is  measured.  The  altitudes 
must  be  corrected  for  parallax  and  refraction,  thus  obtaining  the 
true  altitudes. 

The  annexed  figure  is  a  general  representation 
of  the  triangles  pertaining  to  a  lunar  observation. 
Let  Z  be  the  zenith  of  an  observer,  S'  the 
apparent  place  of  the  sun  or  star,  and  S  its  true 
place.  Also,  let  m'  be  the  apparent  place  of  the 
moon,  and  m  its  true  place  as  seen  from  the  cen- 
ter of  the  earth. 

Here  are  two  distinct  triangles,  ZS'm',  and  ZSm.  The  apparent 
altitudes  subtracted  from  90°,  give  ZS'  and  Zm',  and  S'm'  is  the 
apparent  distance ;  with  these  three  sides,  the  angle  Z  can  be  found. 
Correcting  the  altitudes,  and  subtracting  them  from  90°,  will  give 
the  sides  ZS  and  Zm ;  these  two  sides,  and  their  included  angle  at. 
Z,  will  give  the  side  Sm,  which  is  the  true  distance. 

The  definite  true  distance  must  have  a  definite  Greenwich  time, 
which  can  be  readily  found ;  and  this,  compared  with  the  local  time 
deduced  from  an  altitude  of  the  sun,  will  of  course  give  the  longitude. 
We  shall  now  make  a  formula  to  clear  the  distance. 

Let  £'=the  apparent  altitude  of  the  sun  or  star, 

and  $=the  true  altitude.     Also, 

Let  m'=the  apparent  altitude  of  the  moon, 

and  m  =the  true  altitude. 

Observe  that  the  letters  with  the  accent,  indicate  apparent,  and 
without  the  accent,  the  true  altitudes. 

Put  d  to  represent  the  apparent  distance,  and  x  to  represent  the 
true  distance. 

Bear  hi  mind,  that  the  sine  of  an  altitude  is  the  same  as  the  cosine 
of  its  zenith  distance,  and  conversely,  the  sine  of  a  zenith  distance 
is  the  same  thing  as  the  cosine  of  the  corresponding  altitude. 

Now,  by  the  fundamental  equation  of  spherical  trigonometry  noted 
in,  the  last  chapter,  we  have 


LUNAR   OBSERVATIONS.  223 

„.      cos.  d — sin.  S'  sin.m'      .,  ~_cos.  x — sin.  S  sin.  m 

COS.  J&=  - — .      A1SO  COS.  &— ,  . 

cos.  S  cos.  m  cos.  S  cos.  m 

Whence  COS'  ^ — sin.  S  sin,  m'_cos.  x — sin.  S  sin,  m 

cos.  S'  cos.  m'  cos.  S  cos.  m 

By  adding  unity  to  each  member  we  have 

,    ,cos.  d — sin.  S'  sin.  TO' ,   ,  cos.  x — sin.  S  sin.  m 

cos.  S'  cos.  TO'  cos.  S  cos.  m 

(cos.  S'  cos.  m' — sin.  S'  sin.  m')-J-cos.  d (cos.  S  cos.  m — sin.  Sf  sin.  m)+cos.  *• 

cos.  S'  cos.  m'  cos.  S  cos.  m 

By  observing  equation  9,  plane  trigonometry,  we  perceive  that  the  preceding 
equation  reduces  to 

cos.  (S'-\-m')-\-cos.  d  ^cos.  (S-J-m)-j-  cos.  x 
cos.  S'  cos.  m'  cos.  S  cos.  m 

Whence  cos.  a^=(cos.(fir-{-m')-|-cos.d)  cos>  S  cos  m  —cos.  (SM-m). 

COS.  S'  COS.  TO' 

It  is  here  important  to  notice  that  the  moon's  horizontal  parallax 
given  in  the  Nautical  Almanac,  is  the  equatorial  horizontal  parallax; 
that  is,  it  corresponds  to  the  greatest  radius  of  the  earth.  The 
diameter  of  the  earth  through  any  other  latitude  is  less,  and  of 
course  the  corresponding  parallax  is  less. 

We  therefore  give  the  following  table  for  the  reduction  of  the 
equatorial  horizontal  parallax,  to  the  horizontal  parallax  of  any 
other  latitude  ;  it  is  computed  on  the  supposition  that  the  equatorial 
diameter  is  to  its  polar  as  230  to  229.  For  example  if  the  horizon- 
tal parallax  in  the  Nautical  Almanac  is  55' — ,  hi  the  latitude  of  40° 
the  reduction  would  be  6",  and  the  parallax  reduced  would  be 
54'  54",  and  if  the  parallax  from  the  Nautical  Almanac  were  60' 
the  reduction  would  be  6"  6,  and  reduced  would  be  59X  53' 'A. 

The  semi-diameter  of  the  moon  given  in  the  Nautical  Almanac  is 
her  horizontal  semi-diameter,  but  when  she  is  hi  the  zenith  she  is 
nearer  to  us  by  the  whole  radius  of  the  earth,  about  one-sixtieth 
part  of  her  whole  distance,  consequently  she  must  appear  under  a 
larger  and  larger  angle  as  she  rises  from  the  horizon,  and  this  is 
called  the  augmentation  of  the  semi-diameter. 

We  give  the  reduction  for  the  parallax  ;  and  the  augmentation  for 
the  semi-diameter  in  the  following  tables  : 


NAVIGATION. 


Red.  of 

Lat- 
itude. 

H  's  Eq.  hor.  parallax. 

Ea.r. 

Eq.par. 
60' 

20° 

0".9 

1" 

25 

2  .8 

3 

30 

3  .7 

4 

35 

4  .6 

5 

40 

6  .0 

6  .6 

45 

7  .3 

8 

50 

8  .6 

9  .4 

55 

10  .1 

11 

60 

11 

12 

65 

11  .8 

13 

70 

12  .8 

14 

75 

13  .9 

15 

80 

14  .6 

16 

Augmentation  of  the 
Moon's  semi-diam. 

Ap.  Alt. 

Aug. 

6° 

2" 

12 

3 

18 

5 

24 

6 

30 

8 

36 

9 

42 

11 

48 

12 

54 

13 

60 

14 

66 

15 

72 

16 

90 

16 

We  now  give  an  example  showing  all  the  details  of  finding  the 
longitude  by  a  lunar  observation. 

EXAMPLE. 

Suppose  that  on  the  25th  of  January,  1 852,  between  three  and  four 
o'clock  hi  the  afternoon,  local  time,  the  observed  distance  between 
the  nearest  limbs  of  the  sun  and  moon  was  50°  3'  20",  the  altitude 
of  the  sun's  lower  limb  was  20°  1',  and  of  the  moon's  lower  limb 
48°  57',  height  of  the  eye  16  feet.  The  latitude  corrected  for  the 
run  from  noon  was  34°  12'  IT.,  and  the  supposed  longitude  about 
65°  west.  What  was  the  longitude  ?  (the  Nautical  Almanac  being 
at  hand.) 

Preparation. 


Supposed  time  at  ship, 
Supposed  longitude  65 
Supposed  time  at  Greenwich, 


n.  M. 

3  15  P.M. 

4  20 


7  25  P.  M. 

On  the  25th  at  noon  the  N.  A.  gives  the  f)'s  S.  D.  at  14'  47",  and  at  midnight 
at  14'  45"  7  ;  therefore  at  the  time  of  observation  we  take  Hat  14' 46'',  by  simple 
inspection.  In  the  same  summary  manner  we  take  the  Equatorial  horizontal 
parallax  at  54'  12". 


£)'s  semi-diameter,    - 

* 

14'  46" 

€)'s 

Eq. 

hor.  par. 

- 

. 

54' 

12* 

Aug  for  Alt.  - 

- 

• 

12 

Red.  for  lat.  - 

4 

f)'s  true  S.  D.  - 

. 

14'  58" 

Reduced 

hor.  par. 

. 

,. 

54' 

"IF 

Observed  distance,     - 

500 

3' 

20" 

Alt 

f)'s  L  L        480  5r 

Sun's  S.D. 

16 

16 

€)'* 

S.D. 

14' 

58" 

Moon's  S.  D. 

• 

14 

58 

Dip 

-  3 

56 

Apparent  central  dis. 

500  34'  34"=d 

O'B 

app.  alt. 

49° 

8' 

2"= 

sm.' 

LUNAR   OBSERVATIONS. 


225 


Alt.  Q's  lower  L      20°      1' 
Semi-diameter,               -f-16'  16" 
Dip,                                —  3'  56" 

N.  B.  To  find  the  moon's  parallax  in 
altitude  see  problems  on  page  201. 

f)'s  app.  Alt.  49°  8'           oca.  9.815778 
54'  8"=3248'                     log.  3.511616 
36'  2"=2120                            3.326394. 

49°    8'    2" 
35'  20 
—49 

49°  42'  35"=m 
(S-fm)=69°  53'  53". 

O'sapp.  alt.              20°     13'  12"=S' 
Refraction,                      —  2'  34" 

O'«  true  alt.              20°    10'  38"  S. 
€)'s  app.  alt. 
Parallax  in  alt. 
Refraction, 
True  alt. 
(S'-f  m')=69°  21'  14" 

We  are  now  prepared  to  apply  the  equation  to  compute  the  true 
distance.  The  equation  requires  the  use  of  natural  sines  and 
cosines. 

cos.  S  cos.  m 


cos.  aj=(cos.  (^'-f-m')+cos.  d) 


-cos.  (S+m) 


cos.  S'  cos.  m' 
(S'-H»')=69021'14"N.cos.    .35259 
c*=50°  34'  34"  N.  cos.*  .63449 

.98708  log. —1.994350 
£=20°  10'  38"  log.  cos.  9.972496 
m=49°  43'  15"  log.  cos.  9.810578 
£'=20°  13'  12"  cos.  com.  0.017626 
m'=49°  8'  2"  cos.  com.  0.184228 
Num.  .97561  log.— 1.989278 

=sum  less  20.f 

N.  cos.  (S4-m)=69°  53'  53"  —.34369 
True  distance,      50°  48'  29"  cos.63192 

In  the  Nautical  Almanac,  we  find  that  at  6  P.  M.  mean  Green- 
wich time,  on  said  day,  the  true  distance  between  the  sun  and  moon 
was  49°  69'  26",  and  at  9  P.  M.,  the  distance  was  51°  20'  49",  show- 
ing a  change  of  1°  21'  23"  in  three  hours  of  time.  But  the  change 

*  When  d  is  greater  than  90°  its  cosine  becomes  minus,  and  its  numerical 
Yalue  is  then  the  natural  sine  of  the  excess  over  90°.  Thus  if  d  were  105°,  its 
cosine  would  be  numerically  equal  to  the  sine  of  15°,  and  must  then  be  subtracted 
from  the  cosine  of  the  sum  of  apparent  altitudes.  The  result  (cos.  *)  would 
then  be  the  sine  of  the  excess  over  90°. 

t  Less  20  because  the  table  of  natural  sines  is  to  radius  unity,  and  we  used 
cos.  S  and  cos.  m  to  the  radius  of  10,  making  two  tens  to  take  away. 
15 


226  NAVIGATION. 

from  49°  59'  26"  to  60°  48'  29"  is  49'  3"  ;  and  now  on  the  suppo- 
sition that  the  change  is  in  proportion  to  the  time  (  and  it  is  very 
nearly  ),  we  have  the  following  analogy 

1°  21'  23"  :  49'  3"  :  :  3h.  :  t 

Or,  4883  :  2943  :  :  3  :  \h.  48m.  29s. 

That  is,  the  time  that  this  observation  was  taken  Ih  48m  29s  after 
6  at  Greenwich.  Or,  7h  48m  29s  mean  Greenwich  time. 

With  the  true  altitude  of  the  sun  20°  10'  38",  the  latitude  34° 
12',  and  the  polar  distance  109°  0'  48",  we  find  the  apparent  time 
at  ship  3h  10m  5s,  to  which  we  would  add  the  equation  of  time, 
12m  34,  making  the  mean  time  3h  22m  39s. 

From  the  Greenwich  time        Ih.  48m.  29s. 

Sub.  time  at  ship  -   3     22      39 

Giving  Ion.  in  time  4     25      50=66°  27'  38"  W. 

West,  because  the  time  at  Greenwich  was  later  in  the  day. 

If  a  lunar  is  taken  with  a  star,  or  with  the  sun,  when  the  sun  is  not 
in  a  proper  position  to  depend  upon  its  altitude  for  local  time,  the 
time  must  be  noted  by  a  watch,  and  the  difference  between  the  watch 
and  true  time  made  known,  by  a  previous  or  subsequent  observation 
on  the  sun,  or  some  star  which  is  nearly  east  or  west  of  the  observer. 

The  most  material  part  of  working  a  lunar  is  that  of  clearing  the 
distance.  We,  therefore,  give  the  following  examples,  without  the 
little  incidental  details. 

We  show  the  working  of  one  hi  which  the  distance  is  greater 
than  90°. 

The  apparent  distance  between  the  center  of  the  sun  and  moon 
on  a  certain  occasion,  was  98°  12' ;  the  apparent  altitude  of  the 
sun's  center  was  54°  10',  and  of  the  moon's  20°  37' ;  the  moon's 
horizontal  parallax  at  the  same  time  was  57'  12".  What  was  the 
true  distance  ?  Ans.  97°  25'  10" 

Horizontal  par.  57'  12"=3432  log.     3.532547 
}  Alt.  20°  37'  cos.     9.971256 

Parallax  in  alt.  53°  31'=3211  log.     3.586803 


LUNAR    OBSERVATIONS. 


227 


Q)'s  app.  alt.  20°  37' 
Refraction  —2'  31* 

Parallax  +53'  31" 


^  app.  alt.  54°  10'    0" 
Refraction  — 40" 


'B  true  alt.  54°    9' 20" 


D's  true  alt.  21°  28' 

(^f'_f_m')=74°  47'          (S+7w)=75°  37'  20" 
Nat.  cosine  74°  47'  is     +.26247 
d==98°  12'  Nat.  cos.  is    —14263 


Algebraic  sum  .11984 

#=54°    9'  20"  cos. 
m=21°  28'         cos.     - 
/S"=54°  10'         cos.  complement 
f»'=20°  37'         cos.  complement 


log.  —1.078602 
9.767591 

-  9.968777 
0.232525 

-  0.028744 


Sum  (rejecting  20)         —1.076239 
This  log.  corresponds  to  -{-.11919 

—Nat.  cos.  (,S4-m)=cos.  75°  37'  20"  —.24832 

cos.a?=cos.  97°  25'  10"  —.12913 

N.  B.  The  last  Nat.  cosine  having  the  minus  sign  shows  that 
the  corresponding  arc  must  be  greater  than  90°.  To  find  the 
arc  we  conceived  .12913  to  be  plus,  and  found  it  corresponded 
in  the  table  to  the  Natural  sine  of  9°  25'  10",  and  to  this  we 
added  90°  for  the  result. 

EXAMPLES     FOR     PRACTICE. 


No. 

Ap.  alt.  of  sun 
or  a  fixed  star. 

Moon's  ap.     Apparent  cen- 
altitude.       tral  distance. 

Moon's 
hor.  par. 

True 
distances. 

0           / 

0           / 

O          1         II 

/ 

o       /      // 

1 

O  86    3 

39  18 

46  45    0 

53  51 

46    425 

2 

#  29  47 

5722 

2735    0 

60    3 

28    8  24 

3 

O  31  14 

28    7 

14  21  30 

5429 

14    9  24 

4 

O  60    5 

63  12 

51     3  21 

58  30 

50  41  15 

5 

#  3428 

1042 

49  18  38 

61  11 

48  45  39 

6 

O     8  26 

19  24 

120  18  46 

57  14 

120     1  46 

7 

*  43  27 

40    9 

18  21  35 

60  20 

18    8  12 

8 

*  53  13 

57  32 

60  13  49 

60  52 

59  48  12 

9 

O  72  26 

18  30 

81     228 

6058 

80    933 

10 

O  6033 

9  26 

70  36  16 

59  57 

69  49  12 

228  NAVIGATION. 

The  foregoing  method  of  clearing  a  lunar  distance  is  very 
good,  as  an  educational  exercise,  but  for  practical  use,  it  is  ob- 
jectionable, as  the  equation  requires  the  use  of  natural  sines  and 
cosines.  To  ensure  a  complete  understanding  of  this  important 
subject,  theoretically  and  practically,  we  will  further  transform 
the  equation 

Cm 


cos.  «=(cos./Sf/+w/+cos.J)--:—  cos.rS+w),  (1  ) 
.    coa.S'  cos.m' 

and  adapt  it  to  the  use  of  logarithmic  sines  and  cosines. 

Conceiving  (/S"-j-m')  to  be  a  single  arc,  and  applying  equation 
(17)  (page  50),  to  the  first  factor  in  the  second  member  of  (1), 
we  shall  have 


s.ff  cos.m     CQg  (g  .      } 
cos.S'  cos.m' 

By  equation  (32)  (p.  51),  we  find  that    cos.#=l  —  2sin.2  \x. 
By  eq.  (31)  (p.  51),          cos.(^4-m)==2cos.24(^+m)~  1. 
These  values  of  coa.x  and  cos.(£-|-m),  placed  in  (2)  will  give 

£'4-m'  —  c?)cos.£  cos.m 
cos.  S  cos.m' 


By  dropping  the  units  in  each  member,  and  dividing  by  —  2, 
we  have 
sin.a£p=  (3) 


coB.      cos.m 
By  division,  we  obtain 

sin.2  \x 
cos.3  \(S-\-nC)~ 

S.  S 


cos.  mooB.      cos.m 

Assume 

cos.5cos.m 


cos.2  ^(/S'+mcos./^'  cos.m' 
Calling  P  an  auxiliary  arc,  equation  (4)  now  becomes 


LUNAR   OBSERVATIONS.  229 

Because  Bin.*P-{-coB.aP=lt        cos.aP=l — sin.aP. 

Whence  ™''**      =cos.'P. 

cos.2  £(  8-{-m) 

By  extracting  square  root  and  clearing  of  fractions,  we  have 

sin.i*=cos.P  cos.i(#+m).  (6) 

Equations  (5)  and  (6)  are  plain  and  practical,  they  can  be 
easily  remembered,  and  they  are  adapted  to  logarithms. 

Equations  (5)  and  (6)  can  be  put  in  words,  and  called  a  rule, 
but  in  our  opinion  this  is  not  necessary. 

We  will  now  re-compute  the  last  example,  in  which 
£'=54°  10',     £=54°  9'  20",     m'=200  37',     m=21°  28', 

c?=98°  22',  (£'+m')=74°  47'. 

i(S"+m'+d)=86°  29'  30"     cos.  (less  10)     —2.786704 

|(S'4-»i'- rf)=ll°37'30"    cos.         "  —1.991000 

£=54°    9'  20"    cos.         "  —1.767598 

m=21°28'  cos.  —1.968777 

i(S-\-m)=37°  48'  40"    cos.  complement      0.102364 

*cos.  complement       0.102364 

#'=54°  10'  cos.  complement       0.232525 

*w'=20°  37'  cos.  complement      0.028744 


2)— 2.980076 

—1.490038 
Add  10 


sin.P  18°  0'  7"  9.490038 

cos.P  9.978203 

9.897636 


sin.|*=  9.875838=48°  42'  25" 

2 


True  distance,  97°  24'  50" 

The  two  methods  do  not  give  the  same  result  precisely.     But 
this  one  is  the  most  reliable  of  the  two. 

*  The  preceding  log.  repeated  to  obtain  the  square  of  the  last  quantity. 


APPENDIX. 


IT  is  comparatively  an  easy  matter,  to  conduct  a  survey,  or  navigate 
a  vessel,  when  there  are  no  important  difficulties  to  be  overcome  ;  but 
the  true  test  of  knowledge  or  skill  in  any  pursuit,  is  to  be  found  only  in 
real  adversity.  The  mariner  who  successfully  manages  his  ship,  when 
every  thing  is  provided,  when  all  is  in  order,  and  the  weather  favorable, 
is  deserving  of  little  credit ;  but  let  the  ship,  become  disabled,  and  the 
storm  terrific,  and  then  there  is  scope  for  the  exercise  of  every  neces- 
sary  acquirement,  and  its  kindred  talent. 

So  it  is  with  the  man  of  science  ;  when  every  instrument  is  at  hand, 
and  all  in  order,  it  requires  little  skill,  and  but  common  knowledge,  to 
make  observations  and  experiments  ;  but  when  we  reverse  the  case;  the 
tact,  knowledge,  and  ingenuity  of  the  man,  may  oft  times  more  or  less 
overcome  the  difficulties. 

For  instance,  suppose  it  were  necessary  to  find  the  altitude  of  the  sun, 
for  the  purpose  of  finding  the  latitude  of  the  place  (on  shore),  or  for  the 
purpose  of  finding  the  time ;  and  we  had  no  sextant  or  quadrant,  and  in 
fact,  no  instrument  to  measure  angles.  It  could  be  done  approximately 
as  follows : 

Let  a  plumb  line  be  suspended  in  water  ;  beve  a  knot  in  the  line,  and 
let  the  knot  be  at  a  known  distance  above  the  water.  The  knot  will 
cast  a  shadow  on  the  water  ;  measure  the  distance  of  this  shadow  from 
the  plumb  line.  The  knot  and  its  shadow,  with  the  plumb  line  and 
water,  will  form  a  right  angled  triangle,  and  the  angle  at  the  base,  com- 
puted by  plane  trigonometry,  will  be  the  altitude  of  the  sun's  upper  limb, 
and  this  altitude  may  be  used  for  any  purpose,  the  same  as  if  it  were 
measured  by  a  sextant,  but  the  accuracy  is  not  to  be  depended  upon,  for 
the  want  of  delicacy  in  the  instrument. 

A  person  on  shore  having  a  good  watch,  and  knowing  his  latitude, 
can  regulate  his  watch,  or  at  least  determine  its  rate  and  error  for  a 
short  period  of  time.  Then,  if  he  have  a  nautical  almanac,  the  common 
tables  of  logarithms,  and  a  knowledge  of  spherical  trigonometry,  and  a 


APPENDIX.  231 

corresponding  knowledge  of  astronomy,  he  can  find  the  longitude  by  a 
lunar  observation,  without  a  sextant,  as  follows: 

By  the  means  of  his  watch  and  a  plumb  line,  he  will  be  able  to  range 
off  an  approximate  meridian  line.  He  will  then  observe  the  transits  of 
stars,  and  of  the  moon  across  that  meridian,  taking  those  stars  which 
are  at  that  time  near  the  moon's  meridian,  some  to  the  east  and  some  to 
the  west  of  the  moon,  and  some  more  north,  and  others  more  south 
than  the  moon. 

He  will  note  the  difference  in  time,  between  the  transit  of  each  star 
and  the  moon,  across  his  approximate  meridian,  and  by  a  combination, 
or  rather  comparison  of  these  observations,  he  will  be  able  to  determine 
the  moon's  right  ascension  very  nearly.  By  the  moon's  right  ascen- 
sion, and  the  aid  of  the  nautical  almanac,  he  can  find  the  Greenwich 
time. 

The  Greenwich  time,  compared  with  the  local  time,  will  give  the 
longitude. 

When  we  can  find  the  moon  in  a  vertical  plane  with  any  two  fixed 
stars,  and  it  be  at  the  time  the  moon  changes  her  declination  very 
slowly,  so  that  we  can  depend  upon  a  declination  taken  from  the  nauti- 
cal almanac  for  the  supposed  time,  we  can  then  determine  the  moon's 
right  ascension,  and  from  thence  the  longitude  as  before,  whether  we 
are  on  land  or  sea. 

Ship-wrecked  mariners,  and  travelers  similarly  situated,  have  fre- 
quently resorted  to  these  artifices  to  obtain  their  approximate  localities. 

We  have  frequently  remarked  in  the  course  of  this  work,  that  the 
best  position  of  a  celestial  object,  at  the  time  of  taking  its  altitude,  for 
the  purpose  of  more  exactly  defining  the  time,  is  when  the  object  is 
nearly  east  or  west ;  we  now  propose  to  show  this  conclusively,  and 
therefore  give  the  following 

INVESTIGATION. 

To  find  under  what  circumstances*  in  a  given  latitude,  a  small  mistake  in 
observing  or  correcting  the  altitude  of  a  celestial  object,  witt  produce  the 
smallest  error  in  the  time  computed  from  it, 
Let  Z  be  the  zenith,  P  the  pole,  r  the  supposed  place,  and  m  the  true 

place  of  the  object.     Let  ms  be  a  parallel  of  altitude,  join  the  points  m 

and  r,  and  let  pq  be  the  arc  of  the  equator 

contained  between  the  meridians  Pm  and 

Pr. 

Then  as  Pm  and  Pr  are  equal,  mr  may 
be  considered  as  a  small  portion  of  a  par- 
allel of  declination  rs  will  be  the  error  in 


232  APPENDIX. 

altitude,  and  pq  the  measure  of  the  required  error  in  time.  And  as  the 
sides  of  the  triangle  msr  will  necessarily  be  small,  that  triangle  may  be 
considered  as  a  rectilinear  one,  right  angled  at  s  ;  and  because  the  angle 
Prm  is  also  a  right  angle,  the  angles  smr  and  PrZ,  being  each  the  com- 
plement of  mrs,  are  equal  to  each  other 

We  now  have, 

rs  :  mr  :  :  sin.  smr  (ZrP)  :  rad.  (1) 

Also,  mr  :  pq  :  :  cos.  qr  :  rad.  (2) 

Multiplying  these  two  proportions  together,  omiting  the  common 
factor  mr,  gives, 

rs  :  pq  :  :  cos.  qr  sin.  (ZrP)  :  (rad.)*  (3) 

But,     sin.  rP  or  cos.  qr  :  sin.  rZP  :  sin.  ZP  :  sin.  (ZrP)    (4) 
Whence,  cos.  qr  sin.  ZrP=sin.  rZP  sin.  ZP  (5) 

The  first  member  of  equation  (5),  is  the  same  as  the  third  term  in 
proportion  (3)  ;  therefore,  proportion  (3)  may  be  changed  to  the 
following, 

n  :pq:  :  sin.  rZP  sin.  ZP  :  (rad.)» 
mence,  ^=  /  »  (rad.)M  _1 

\  sin.  ZP  /  sin.  rZP 
Now,  as  the  quantities  in  parentheses  are  supposed  to  be  constant  the 

value  of  pq,  the  error  in  time  must  vary  as  —  --  —  _  varies  ;  and  it  is 

sin.  rZP 

obvious  that  pq  will  be  least,  when  sin.  rZP  is  greatest,  that  is,  when 
rZP=90°,  or  the  object  due  east  or  west 

Again,  we  can  come  to  a  like  result,  more  directly  and  ele- 
gantly, by  the  direct  application  of  the  differential  calculus. 

Let  PZr  be  the  spherical  triangle,  from  which  time,  or  the 
angle  ZPr,  is  computed.  This  angle  will  vary  as  the  altitude 
varies,  ZP  and  Pr,  the  co-latitude  and  polar  distance,  being  con- 
stant for  any  small  portion  of  time. 

Let  A  represent  the  sun's  true  altitude,  L  the  latitude  of  the 
observer,  D  the  sun's  polar  distance,  and  P  the  angle  at  the 
pole,  included  between  the  meridian  of  the  observer  and  the 
meridian  of  the  sun. 

Now,  by  a  fundamental  equation  in  spherical  trigonometry, 
we  have 


COB.L  sin.D 

Now  the  altitude  of  the  sun,  A,  varies  every  instant,  and  this 
Foot  of  what  is  now  p.  230  in  Ap.  to  Surv 


APPENDIX.  233 

causes  the  value  of  P  to  vary,  but  L    and  D  are  constants, 
therefore  the  differential  of  the  above  equation  will  be 

TUT*         GOS.AdA  /,v 

— sm.PdP= — . — — .  ( 1 ) 

coB.LBin.J) 

But  in  the  triangle  ZPr  we  have 

coa.A  :  sin.P  :  :  sin.Z)  :  sin.Z. 


sin.Z> 

This  value  of  sin.P  placed  in  (1)  will  give 
cos. A  sin.Z  ,p__    cos.^cL4 
sin.D  cos.Z/  sin.j9 

Reducing,  we  find         —  dP= ~- 

cos.Z  sin.Z 

Now  as  cos.Z  is  a  constant  quantity,  the  value  of  — dP,  or 
the  second  member,  will  be  least  when  sin.Z  is  greatest.  That  is, 
when  Z  is  a  right  angle,  and  the  sun  due  east  or  west  of  the 
observer. 

The  minus  sign  before  dP  shows  that  when  A  increases,  P 
decreases,  which  is  obviously  true  at  all  times. 

When  L—0,  that  is,  the  observer  on  the  equator,  the  result 

will  be  — dP= ,  and,  if  we  suppose  the  sun,  also,  on  the 

sin.Z 

equator,  it  will  all  the  while  be  either  east  or  west  of  the  ob- 
server, and  then  — dP=dA  ;  that  is,  the  time  and  altitude  would 
then  have  equal  variation. 


LUNAR    OBSERVATIONS. 


The  differential  calculus  will  apply  most  beautifully  to  the 
clearing  of  lunar  distances  from  the  effects  of  parallax  and  re- 
fraction. 

In  this  case  we  must  regard  the  difference  between  the  true 
and  apparent  altitude  of  the  moon,  as  a  differential  quantity, 
and  the  refraction  of  the  sun  or  star  a  differential  quantity,  and 
the  difference  between  the  true  and  apparent  distance  is  a  cor- 
rection sought,  and  it  is  also  to  be  regarded  as  a  differential 
quantity. 


234  APPENDIX. 

Let  ZS'm'  represent  the  observed  triangle. 
The  observed  center  of  the  sun  or  star  is  at 
S',  but  the  center  really  is  at  S.  The  ob- 
served center  of  the  moon  is  at  m',  the  real 
center  i  at  m,  as  seen  from  the  center  of  the 
earth.  S  'm'  is  the  apparent  distance,  but  the 
true  distance  is  Sm. 

Let  S=  the  apparent  altitude  o    the  sun  or  star. 
m==  the  apparent  altitude  of  the  moon. 

And#=  the  apparent  or  observed  distance  S'm'. 

Now  by  spherical  trigonometry,  (see  page  223  of  this  book,) 
we  have 


cos  Z=  cos'x  —  8*n-S  sin.m  ,  ^  * 

cos.  S  cos.  m 

In  this  problem  the  angle  Z  is  always  a  constant  quantity,  S 
and  m  are  variable,  and  x  varies  in  consequence  of  the  variations 
of  S  and  m.  But  we  may  take  these  effects  separately.  That 
is,  by  supposing  m  only  to  vary,  and  discover  the  corresponding 
variation  for  x.  Then  we  may  suppose  £  to  vary,  and  obtain  the 
corresponding  variation  to  x  ;  and  lastly,  these  two  effects  put 
together  will  be  the  total  variation  for  x,  or  the  difference  between 
the  apparent  and  true  distance  between  the  sun  and  the  moon 
or  a  star  and  the  moon,  as  the  case  may  be. 

We  will  therefore  differentiate  (1)  on  the  supposition  that  x 
and  m  are  variables. 

That  is,  d.     cos.Z  cos.  S  cos.m—  d.  cos.x  —  c?.  sin.  S  sin.m. 

Or  —  cos.Z  cos.  S  sin.m  dm=  —  sin.a;  dx  —  sin.  5  cos.m  dm* 

sin.x  —  =cos.Z  cos.  S  sin.m  —  sin.  S  cos.m.         (2) 
dm 

T»   ,  .•      /+\     •  rr         n    cos.  a:  —  sin.  S  sin.  m 

But  equation  (  1  )  gives       cos.Z  cos.  S==  ------- 

cos.m 

Multiply  by  sin.m,  then 

f7         e   •  cos.  x  sin.m  —  sm.£sm.2m 

cos.Z  cos.  o  sm.m=  -  . 

cos.m 

This  value  placed  in  equation  (2),  that  equation  becomes 

dx     cos.  x  sin.m  —  sin.«S>  sin.2m       •     e^ 
8in.#  _  —  .  —  sin.o  cos.m. 

dm  cos.m 


APPENDIX.  235 

Or    co  s  .m  BITL.X  —  =cos.#  sin.m  —  sin.  S  sin.a  m  —  sin.  S  cos.  2m. 
dm 

=cos.a;  sin.m  —  sin.£(sin.2m-j-cos.am). 
But  (sin.2m-|-co8.2m)=l.    Therefore 


dx 

cos.wsm.a;  —  =cos.a:  sin.m  —  am.S. 
dm 


Or 


cos.w  sn.a: 


Now  if  we  suppose  that  x  and  S  are  the  variables,  in  place  of 
x  and  m,  the  result  will  be  the  same  as  (3)  if  we  change  S  to  m 
and  m  to  & 


Therefore  ..—.^  ig  the  yalue  of  ^  corres. 

\        cos.  S  sin.a;      / 
ponding  to  the  variation  of  the  sun  or  star's  altitude. 

The  apparent  place  of  the  moon  is  below  its  true  place,  and 
the  apparent  place  of  the  sun  or  star  is  above  its  true  place, 
therefore  dm  and  dS  must  have  contrary  signs.  Consequently, 
the  whole  variation  of  a;,  when  both  S  and  m  vary,  (as  is  always 
the  case,)  must  be 

,  _  /cos.a:  sin.m  —  sin./S\  ,   ___/cos.arsin./Sf  —  sin.m\  ,«    /4\ 
\       cos.m  sin.a;        /          \        cos.  S  sin.a:       / 

When  the  sun  or  star  is  at  the  zenith,  (dS)  is  then  nothing, 
and  the  value  of  dx  is  expressed  by  the  first  term  of  the  second 
member.  When  the  moon  is  in  the  zenith,  then  (dm)  becomes 
nothing  ;  but  in  practice,  such  cases  would  not  be  likely  to  occur, 
once  in  a  life  time. 

We  will  work  the  fourth  example  by  this  formula  : 

Given,  the  sun's  apparent  altitude  60°  5'.  The  moon's  apparent 
altitude  63°  12'.  The  apparent  distance  51°  3'  21",  and  the  moon's 
horizontal  parallax  58'  30",  to  find  the  true  distance  from  center  to 
venter,  as  seen  from  the  center  of  the  earth.* 

Ans.  60°  41'  15". 

*  Correction  may  be  made  for  the  figure  of  the  earth  by  correcting  the 
parallax  for  the  latitude  of  the  observer,  as  shown  in  table  on  page  224. 


236  APPENDIX. 

Here      8=60°  5',  w=63°  12',  *=51°  3'  21". 

Q)  h.  p.=58'  30"=3510*,     log.     - 


cos.  m 


Parallax  in  altitude    +1582" 
Refraction  —29" 


3.645307 
9.654059 

3.199366 


+ 1 563"         dS=  — 32"  sun '  s  re  fraction 


For  the  coefficient  of  dm, 

sin.m  — 1.960650 
— 1.798351 


cos.m  — 1.654059 
— 1.890843 


.56108  —1.749001 


N.  sin.tf  —  .86675 


—.30567  log. 

dm  1553  log. 
First  part  of  dx  —1354", 


—1.644902  Den. 


-      -*-1.486266  Num. 

—1.940363 
•*          3.191171 


3.131534 


For  the  coefficient  of  dS, 

sin.£  —1.937895 
cos.«  —1.798351 


cos.S  —1.697874 
sin.a:    — 1.890843 


+.54480     —1.736246 
N.  sin.m — .89259 


—1.588717  Den. 


—.34779  log. 


-        -      —1.540078  Num. 

—  1.951361 
rf#=—  32"     log.     +1.505150 

28"6 

Whence    dx——  1364'+28"6=  —22'    6" 
Apparent  distance,          51°    3'  21" 

True  distance,  60°  41'  15" 


1.456511 


The  equation 

,  ^_(cos.#  sin.m — sin./S)  ,        /cos.x  sin./Sf — sin.m\  ,« 

cos.m  sin.a;  \       cos.  S  sin.*        / 

can  be  put  into  another  form,  which  will  better  suit  the  tastes 
of  mere  practical  men,  and  avoid  the  use  of  natural  sines. 


APPENDIX.  237 

Assume    cos.a?  sin.m=sin.^,    and    COB.X  sin.5=sin.jB,    and 
determine  the  values  of  A  and  JB.    Then  we  shall  have 
dx__  (sin.  A  —sin.  S)^     /sin.£—  sin. 
cos.msin.a;  \   cos./S'sin.a: 

The  first  term  of  the  second  member  will  be  plus  or  minus, 
according  as  A  is  greater  or  less  than  8.  The  coefficient  of  dS 
is  positive,  when  B  is  greater  than  m.  But  (dS)  is  always  nega- 
tive ;  hence,  the  product  will  be  positive  or  negative,  according 
to  the  rules  of  algebraic  multiplication. 

By  the  application  of  equation  (16),  page  50,  and  dividing  bj 
2,  the  preceding  equation  becomes 
|<fc  = 

S)  ^  _ 


cos.m  sin.ic  cos.  S  sin.a; 

This  equation,  put  in  words,  would  be  a  rule  for  clearing  the  dis- 
tance, but  those  who  comprehend  it,  can  follow  it  as  readily  with- 
out the  words,  as  with  them.  We  illustrate  by  a  single  example. 

The  sign  u/5  indicates  the  difference  between  the  two  quantities  between 
which  it  is  placed,  when  it  is  not  known  which  is  the  greatest. 

EXAMPLE. 

Suppose  that  in  latitude  46°  north,  the  moon's  apparent  altitude 
was  36°  28',  and  that  of  a  planet  24°  43',  and  their  apparent  dis- 
tance asunder  was  71°  46'  24".  The  moon's  horizontal  parallax  at 
that  time  was  58'  31",  and  that  of  the  planet  29";  what  was  the  trw 
distance  as  seen  from  the  center  of  the  earth? 

PREPARATION. 

Q)'s  tor-  par.     58'  31* 
Table,  page  224,  7"7 


58'  23"3=3503.3     log.  3.544477 
cos.Q)'s  alt.  36°  28'  9.905366 


Q)'s  parallax  in  alt.  2817"          3.449843 
Refraction        -        —77 


238 


APPENDIX. 


Planefc. 

Log.  29"     1.462398 
Cos.  24°  43'     9.958271 


Parallax  in  Alt. 
Refraction,         • 


+26"3 
—124 


1.420669 


dS  —  97"7 

Here  m=36°  28',     #=24°  43',     and    #=71°  46'.* 
For  the  auxiliary  arcs  A  and  B9 

cos.*  71°  46'     9.495388         -         *         -         9.495388 
sin.m  36°  28'     9.774046  sin.tf  24°  43'     9.621313 


A 
S 


10°  43'     9.269434 
24°  43' 


m 


7°  31'     9.116701 
6°  28' 


1=17°  43' 
|=  7°    0' 

We  now  follow  the  formula. 

1st  Term. 

C08.%(S+A)  17°  43'  9.978898 
Bin4(S—A)  7°  0'  9.085894 
cos.  compl.wi  36°  28'  0.094634 
sin.  comply  71d  24'  0.023298 
dm  2740"  3.437751 


(Less  20).  — 417"3      2.620475 


=21°  59' 
J?)=14°28' 


2nd  Term. 

21°  59'  9.967217 
14°  28'  9.397621 
cos.  comple.     24°  43'  0.041729 
sin.  comple.     71°  24'  0.023298 
dS  —  97"7       1.989895 


+26"29     1.419760 


The  first  term  is  minus,  because  A  is  less  than  S,  and  the 
second  term  is  plus,  because  it  contains  the  product  of  two 
minus  factors,  (sin.jB — sin.m)  and  dS. 

Whence        !<&=— 41 7.3+26"3=— 39 1" 

Or  efe=— 782"=  —13'    2" 

Apparent  distance,  71°    48'  24" 


True  distance, 


-     71°    33'  22' 


•  In  computing  the  coefficients,  seconds  of  arc  need  not  be  noticed. 


APPENDIX. 


239 


We  give  the  following  examples  of  distances  between  the 
moon  and  planets,  for  exercises : 


No. 

Moon's  Appa- 
rent Altitude. 

Planet's  Ap- 
parent Alt. 

Moon's  Dist. 
from  Planet. 

Moon's  Hor. 
Parallax. 

Planet's 
Parallax. 

Tn» 

Distance. 

1 

58     36 

16     23 

69  37  20 

56       0 

31 

69  40  30 

2 

80       4 

35     30 

60     4     3 

61     16 

18 

59  58  57 

3 

16     26 

29     41 

98  15  31 

60     35 

30 

97  45     4 

4 

50     14 

51       3 

40     0     0 

54     50 

25 

39  44  42 

5 

62     12  i 

38     27 

37  50  34 

55     13 

23 

37  58  14 

LOGARITHMS. 

In  the  forepart  of  this  volume,  we  have  shown  the  practical 
uses  of  logarithms,  using  the  common  tables,  which  extend  only 
to  six  places  of  decimals,  and  this  is  sufficient  for  all  common 
purposes.  But  for  those  who  desire  to  be  more  nice  and  accu- 
rate, we  have  computed  a  table  extending  to  twelve  decimal 
places,  including  the  consecutive  numbers  from  1  to  200,  and 
from  thence,  the  prime  numbers  as  far  as  1543,  together  with 
the  Auxiliary  Logarithms  of  unity,  and  a  fraction  as  low  as  the 
number  1.0000000001. 

In  these  logarithms  the  index  is  omitted,  as  it  is  not  necessary 
when  one  has  obtained  the  true  theory  of  logarithms*  For  in- 
stance, the  log.  of  the  number  28  has  a  certain  decimal  part, 
which  must  remain  the  same  if  that  number  be  changed  to  2.8, 
or  to  .28,  or  to  280,  &c.  &c.,  and  according  to  the  value  of  the 
2  and  8,  the  operator  will  prefix  the  index. 

To  make  a  table  of  logarithms  anew,  to  contain  any  particular 
number  of  decimal  places,  the  following  formula,  taken  from 
algebra,  is  the  most  practical  and  convenient  of  any  yet  known. 

Log.(«+l)= 

log.2+.8685889638f- 


\2z+l   '  3(22+1)3  '  5(20+1) 

This  formula  will  give  the  log.  of  (2+!)  when  the  log.  of  z 
is  known,  but  the  log.  of  z  is  known  when  we  make  0=1,  10, 
100,  1000,  <fec.  Then  the  formula  will  give  the  logarithms  of 
2,  11,  101,  1001,  &c. 


240  APPENDIX. 

When  z  is  large,  Gf0f  100,  the  series  converges  very  rapidly, 
and  then,  only  two  terms  need  be  used.  When  z  is  over 
2000,  only  two  terms  need  be  used,  even  for  twelve  decimal 
places. 

The  auxiliary  logarithms,  A,  B,  C,  page  71  of  tables,  were 
computed  by  this  formula.  For  instance,  the  log.  of  1001  is 
the  same  in  its  decimal  part  as  the  log.  of  1.001.  Hence,  when 
we  have  the  logarithms  of  1001,  1002,  1003,  &c.,  we  have  the 
logarithms  of  1.001,  of  1.002,  &c. 

The  greater  the  number  the  more  readily  can  its  logarithm  be 
computed. 

That  the  learner  may  fully  comprehend  the  application  of 
these  auxiliary  logarithms,  A,  J3,  C,  he  must  call  to  mind  the 
following  principle  : 

(ART.  14.)  The  product  of  any  number  of  factors  consisting  of 
unity  and  a  small  fraction,  is  very  nearly  equal  to  unity,  and  the 
sum  of  those  fractions. 

Thus  the  product  of  (1.0001),  (1.00002),  (1.000004),  is  very 
nearly  equal  to  1.000123. 

If  this  be  true,  we  can  immediately  separata  i. 000123  into 
the  same  factors. 

The  number  1.00021  maybe  taken  for  the'prwAuctof  (1.00004) 
(1.00017),  without  any  material  error. 

Or  (1.00021)  maybe  taken  for  the  produoi  of  21  factors,  each 
equal  to  (1.00001),  with  very  little  error;  and  if  this  be  true, 
1.00001  is  the  21st  root  of  1.00021  very  nwrly. 

This  principal  may  be  proved  algebraically  thus:  Let  a,  b, 
and  c,  be  very  small  fractions,  then  the  product  of  ( l-f-«)  •  ( 1+^) 
is  l-{-a-(-5-{-a6,  as  we  find  by  actual  multiplication. 

But  a  and  b  being  very  small  fractions,  their  product  ab  is 
extremely  small  in  respect  to  a  or  b,  and  therefore  ab  may  be 
omitted,  and  the  essential  part  of  the  product  is  l-|-a-|-£,  that  is, 
unity  and  the  sum  of  the  fractions. 

Now  let  (a-\-b)  be  s,  and  the  product  of  (1+*)  into  (1+c) 
will  be  1+s+c  ;  that  is,  the  product  of  (1+a)  (l+b)  (1+c) 
cannot  be  far  from 


APPENDIX.  241 

Let  us  try  this  in  numbers.    Multiply  1.0001  by  1.00004. 

1.00004 
1.0001 


100004 
1.00004 

1.000140004 

But  this  value  is  extremely  near  1.00014,  the  sum  of  unity  and 
the  fractional  parts  of  the  factors. 

(ART.  15.)  When  the  difference  between  two  quantities  of 
the  same  kind  is  very  small  in  relation  to  the  quantities  them- 
selves, such  a  difference  (in  the  higher  mathematics)  is  some- 
times called  a  differential. 

Thus,  in  the  last  example,  the  difference  between  1.000140004 
and  1.00014  is  .000000004,  and  it  is  so  small  in  relation  to 
1.00014  that  it  may  be  omitted  without  sensible  error,  and  it  is 
then  a  practical  differential. 

The  difference  between  8  and  9  is  1,  but  in  this  case  1  cannot 
be  a  differential,  it  is  too  large. 

The  difference  between  80000  and  80001  is  1,  and  here  1 
might  be  taken  as  a  differential  in  some  practical  computations, 
and  therefore  omitted. 

There  is  no  exact  line  of  demarkation  where  a  difference  may 
be  taken  for  a  differential  ;  'that  depends  on  the  nature  of  the 
case  ;  hence,  those  who  distrust  their  own  judgments  are  gener- 
ally prejudiced  against  the  calculus. 

If  we  take  the  logarithmic  formula  from  (Art.  13,)  and  con- 
ceive z  to  be  very  large,  then  the  difference  between  (z-\-\)  and 
z  is  one  (very  small),  and  may  be  regarded  as  the  differential  of 
«:  and  in  that  case  log.(z-|-l)  —  log.z  is  the  same  as  the  differ- 
ential of  the  logarithm  of  z. 

Making  this  supposition,  the  formula  in  (Art.  13)  becomes 

(dif.)  lo 


We  take  but  one  term  of  the  series,  because  the  following 
terms  are  of  no  essential  value  compared  to  the  first,  and  as  z  is 
very  large,  (2^+0  is  comparatively  so  little  greater  than  2z, 
16 


242  APPENDIX. 

that  for  all  practical  purposes,  it  may  be  taken  as  2z,  and  then  the 
preceding  equation  will  reduce  to 

(,if)^,_0.4848944819(dif.),  (J), 

The  symbol  (dif.)  signifies  the  differential  of  the  quantity 
which  follows  it,  and  this  equation,  put  in  words,  gives  the  fol- 
lowing rule  to  adjust  a  logarithm  to  correspond  to  any  particular 
number. 

RULE.  —  The  differential  of  a  logarithm  in  equal  to  the  differential 
of  the  number  multiplied  by  the  modulus,  and  the  product  divided  by 
the  number. 

When  we  wish  to  find  the  differential  of  a  number  corres- 
ponding to  a  given  differential  of  a  logarithm,  we  change  the 
equation  to  the  following  : 


-. 
0.4342944819 

This  gives  the  following  rule  to  correct  a  number  : 

RULE.  —  The  differential  of  a  number  is  equal  to  the  number  mul- 

tiplied into  the  differential  of  the  logarithm,  and  that  product  divided 

by  the  modulus  of  the  system. 

The  practical  use  of  equations  (1)  and  (2)  will  be  found  in 
the  following 

EXAMPLES. 

1  .  When  the  diameter  of  a  circle  is  1  ,  the  circumference  is 
3.14159265359.  Find  the  logarithm  of  this  number  true  to  at 
least  ten  decimal  places. 

The  number  being  between  3  and  4,  the  index  is  0.  During 
the  operation  we  shall  pay  no  regard  to  the  index  of  any  loga- 
rithm we  may  take  out,  because  it  will  not  be  necessary.  We 
may  consider  the  number  to  be  314  and  a  decimal,  or  we  may 
take  the  whole  for  a  whole  number,  but  it  is  best  to  take  314, 

*  This  equation  is  found  in  the  differential  calculus,  thus  :  dx=  —  —  ,  an 

equation  in  which  x  is  a  logarithm,  y  a  number,  and  m  the  modulus.  Here 
then,  rough  and  practical  as  our  equations  appear  to  be,  they  exactly  coin- 
cide with  the  most  refined  theory. 


APPENDIX. 


243 


the  three  superior  digits,  as  a  whole  number,  whatever  the  number 
may  be. 

Table  III.  commencing  on  page  67  of  tables,  and  the  auxiliary 
logarithms  following,  are  the  log.  referred  to. 
314=157X2. 

Log.  157  - 

Log.       2     .. 


0.195899652409 
.  0.301029995664 


Log.  314 
Table  B,  1.0005    log. 


0.496929648073 
0.000217092970 


Product, 
Table  C, 


3141570     log.         -         0.497146741043 
1.000007     log.       0.000003040047 


Product, 
Given  number, 


2 199099 
314157 

314159  199099     log. 
314159265359=0 


0.497149781090 (a) 


Diff.  66260=(dif.)z. 

We  have  log.  (a)  the  logarithm  of  a  number  very  near  z,  so 
near,  that  the  difference  may  be  called  a  differential ;  therefore 
we  may  use  equation  (1). 

(dif.)  Ugr-(0-48429448)(66260) 
314159199099 

But  we  may  take  common  logarithms  to  reduce  this  second 
member.     In  that  case  we  have  only  one  log.  to  find,  as  the  log. 
of  the  modulus  is  constant,  and  (a)  may  be  used  for  log.  of  z. 
log.  66260       -  4.821251 

log.  m         -         -         -         -  —1.637784 


log.  z     - 

Num.  0.000000091596 

To  (a)  - 
Add  dif.  of  z 


4.459035 
11.497150 

—8.961885 
0.497149781090 
0.000000091596 


Num.  3.14159265359    log.  —0.497149872686  Ang. 


244  APPENDIX. 

The  factor  1.0005  was  obvious,  but  the  other  factor  1.000007 
may  not  be  so,  —  the  question  then  is,  how  did  we  obtain  it? 
It  was  thus : 

3.14157a?=3.14159265359( 
Whence  x=  1 .000007+. 

2.     The  sidereal  year  consists  of  365.2563744  mean  solar  days. 
What  is  the  logarithm  of  this  number? 

If  we  take  the  four  superior  figures,  3652,  as  a  whole  number, 
and  factor  it,  we  shall  find  that  it  is  equal  to  44-83.     Hence 
Log.       44      -  0.643452676486 

Log.       83  0.919078092376 

Log.  3652     ....  0.562530768862 

(1.0001         log.  (B)            -  .000043427277 

Factors  Jl.00005      log.  (G)       -        -  21712704 

(1.000004    log.  (0)          -         -  1737173 


365.2562408     log.     ....     2.562597646016  (a) 
365.2563744  given  number. 


.0001336  differential. 

We  obtained  the  above  factors  by  solving  the  following  equa- 
tion : 

3652*=365.2563744. 

Whence  a?=1.000154,  (Art.  14,)  (1.0001)  (1.00005)  (1.000004). 
To  correct  (a),  and  make  it  exactly  correspond  to  the  given 
number,  we  have  the  following  expression  : 

dif.  log  -  (°-43429448)(1336) 
3652562408 

By  common  logarithms,  log.  m        — 1.637784 

log.  1336,         3.125806 

2.763590 
log.  a  9.562597 

Num.  0.000000158500,  _7.200993 

To  (a)         -         -         -         -      2.562597646016 
Diff.         ....  .000000158500 


Log.  sought,       ....     2.562597804516 


APPENDIX. 


245 


3.  When  the  radius  of  a,  circle  is  1,  the  natural  sine  of  7°  30' 
is  expressed  by  the  decimal  0.1305261921.  What  is  the  logarithm 
of  this  number? 

-    0.113943352307 
0.001733712775 


Log.  .13 
Log.      1.004 

Product 


.13052 
f  1.00004         log. 
Factors    3 1.000007      log.  - 
(1.0000004  log. 


0.115677065082 

-    OOOjl 7371430 

3040047 

173716 


4(n) 


Log.  sin.  of  7°  30'   (Rad.  unity,)       —1.1 15697650275 
For  the  common  table,  add        -         10. 


Log.  sin.  7°  30',  for  table,          -  9.115697650275 

By  the  foregoing,  the  reader  will  perceive  that  the  logarithm 

of  any  number  can  be  found  by  these  tables,  true  to  at  least  ten 

places  of  decimals. 

We  are  now  prepared  to  take  the  converse  problem,  that  is  : 

Given  a  logarithm  to  find  its  corresponding  number. 

EXAMPLES. 

1.      What  number  corresponds  to  the  log.  4.636747519487? 

Comparing  the  decimal,  with  the  decimal  logarithms  in  the  ta- 
ble, we  perceive  that  it  corresponds  to  a  number  which  is  a  little 
greater  than  433  ;  but  as  the  index  is  4,  the  real  number  must 
be  a  little  greater  than  43300.  Let  this  be  one  factor  of  the 
required  number,  then 

From  the  given  log.         -         -        4.636747519487 

Subt.  log.  of  43300     -  -    4.636487896353 

0.000259623134 

217092970  next  less 
in  B. 


Log.  1.0005,  table  B, 

Log.  1.00009,  table  (7, 
Log.  1.000007,  table  <7, 


.000042530164 
39083266 


3446898 
3040047 


406851 


246  APPENDIX. 

.000000406851 
Log.  1.0000009,     ....  390861  9(n) 

15990 
Log.  1.00000003        -        -        -  13029  3(0) 

2961 

The  product  of  these  small  factors  is  1.00059793.  (Art.  14.) 
Multiply  this  product  by  the  factor  43300. 

Or  which  is  the  same  thing, 

Multiply  100.059793 

By  433 

300  179379 
3001  79379 
400239172 


Approximate  number,  43325.890369 

But  the  last  remainder  in  the  logarithm  (2961)  may  be  taken 
as  the  differential  of  a  logarithm,  and  corresponding  thereto  is 
a  differential  of  the  number,  which  must  be  added. 

It  is  found  thus : 

Diff.  of  the  number=(43325)('OQOQg°QQgg6l). 

.43429448 

By  log.        log.  43325        -        -    4.636747  (the  given  log.) 
.000000002961     -         -     —9.471438 


—4.108185 
Log.  .43429,  <fec.  —1.637784 

Correction  0.0002954  log.  —4.470401 

Approx.  num.   43325.890369 

Num.  sought,   43325.8906644 

EXAMPLES. 

1.  What  number  corresponds  to  the  log.  2.204923118054? 

Ans.   160.29616. 

2.  What  number  corresponds  to  the  log.  4.133409102? 

Ans.  13595.93. 

3.  What  number  corresponds  to  the  log.  3.2789020746? 

Ans.  1900.64967. 


LOGARITHMIC  TABLES; 


ALSO    A   TABLE    OP    THE 


TRIGONOMETRICAL    LINES; 

AND  OTHER  NECESSARY  TABLES. 


LOGARITHMS    OF   NUMBERS 

FROM 

1    TO    10000, 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0  000000 

26 

1  414973 

51 

1  707570 

76 

1  880814 

2 

0  301030 

27 

1  431364 

52 

1  716003 

77 

1  886491 

3 

0  477121 

28 

447158 

53 

1  724276 

78 

I  892095 

4 

0  602060 

29 

462398 

64 

1  732394 

79 

1  897627 

5 

0  698970 

30 

477121 

55 

1  740363 

80 

1  903090 

6 

0  778151 

31 

491362 

56 

1  748188 

81 

1  908485 

7 

0  845098 

32 

505150 

57 

1  755875 

82 

1  913814 

8 

0  903090 

33 

518514 

58 

1  763428 

83 

1  919078 

9 

0  954243 

34 

1  531479 

59 

1  770852 

84 

1  924279 

10 

1  000000 

35 

1  544068 

60 

1  778151 

85 

1  929419 

11 

041393 

36 

1  556303 

61 

1  785330 

86 

1  934498 

12 

079181 

37 

568202 

62 

1  792392 

87 

1  939519 

13 

113943 

38 

579784 

63 

1  799341 

88 

1  944483 

14 

146128 

39 

591065 

64 

1  806180 

89 

1  949390 

15 

176091 

40 

602060 

65 

1  812913 

90 

1  954243 

16 

204120 

41 

612784 

66 

1  819544 

91 

1  959041 

17 

230449 

42 

623249 

67 

1  826075 

92 

1  963788 

18 

255273 

43 

633468 

68 

1  832509 

93 

1  968483 

19 

278754 

44 

643453 

69 

1  838849 

94 

1  973128 

20 

301030 

45 

653213 

70 

1  846098 

95 

1  977724 

21 

322219 

46 

662578 

71 

1  851258 

96 

1  982271 

22 

342423 

47 

672098 

72 

1  857333 

97 

1  986772 

23 

361728 

48 

681241 

73 

1  863323 

98 

1  991226 

24 

380211 

49 

690196 

74 

1  869232 

99 

1  995635 

25 

397940 

50 

698970 

75 

1  875061 

100 

2  000000 

N.  B.  In  the  following  table,  in  the  last  nine  columns  of  each  page,  where 

the  first  or  leading  figures  change  from  9's  to  O's,  points  or  dots  are  now 

introduced  instead  of  the  O's  through  the  rest  of  the  line,  to  catch  the  eye, 

and  to  indicate  that  from  thence  the  corresponding  natural  numbers  in 

the  first  column  stands  in  the  next  lower  line,  and  its  annexed  first  two 

figures  of  the  Logarithms  in  the  second  column. 

LOGARITHMS  OF  NUMBERS.      3 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

000000 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

3461 

3891 

101 

4321 

4750 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

102 

8600 

90% 

9451 

9876 

.300 

.724 

1147 

1570 

1993 

2415 

103 

012837 

3259 

3680 

4100 

4521 

4940 

5360 

6779 

6197 

6616 

104 

7033 

7451 

7868 

8284 

8700 

9116 

9632 

9947 

.361 

.776 

105 

021189 

1603 

2016 

2428 

2841 

3262 

3664 

4075 

4486 

4896 

106 

5306 

5716 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

107 

9384 

9789 

.195 

.600 

1004 

1408 

1812 

2216 

2619 

3021 

108 

033424 

3826 

4227 

4628 

6029 

5430 

5830 

6230 

6629 

7028 

109 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

.207 

.602 

.998 

110 

041393 

1787 

2182 

2576 

2969 

3362 

3755 

4148 

4640 

4932 

111 

6323 

6714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

112 

9218 

9606 

9993 

.380 

.766 

1153 

1638 

1924 

2309 

2694 

113 

053078 

3463 

3846 

4230 

4613 

4996 

6378 

6760 

6142 

6524 

114 

6906 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

.320 

116 

060698 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

116 

4458 

4832 

5206 

6580 

6953 

6326 

6699 

7071 

7443 

7815 

117 

8186 

8557 

8928 

9298 

9668 

..38 

.407 

.776 

1145 

1514 

118 

071882 

2260 

2617 

2986 

3352 

3718 

4086 

4461 

4816 

5182 

119 

6547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

120 

9181 

9643 

9904 

.266 

.626 

.987 

1347 

1707 

2067 

2426 

121 

082785 

3144 

3603 

3861 

4219 

4576 

4934 

5291 

6647 

6004 

122 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9562 

123 

9905 

.258 

.611 

.963 

1315 

1667 

2018 

2370 

2721 

3071 

124 

093422 

3772 

4122 

4471 

4820 

6169 

6618 

6866 

6215 

6562 

126 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

1026 

126 

100371 

0716 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

127 

3804 

4146 

4487 

4828 

6169 

6510 

5851 

6191 

6531 

6871 

128 

7210 

7649 

7888 

8227 

8666 

8903 

9241 

9679 

9916 

.253 

129 

110590 

0926 

1263 

1699 

1934 

2270 

2605 

2940 

3275 

3609 

130 

3943 

4277 

4611 

4944 

5278 

6611 

6943 

6276 

6608 

6940 

131 

7271 

7603 

7934 

8265 

8696 

8926 

9266 

9586 

9916 

0246 

132 

120674 

0903 

1231 

1660 

1888 

2216 

2544 

2871 

3198 

3525 

133 

3852 

417.8 

4504 

4830 

5156 

6481 

6806 

6131 

6466 

6781 

134 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

..12 

136 

130334 

0656 

0977 

1298 

1619 

1939 

2260 

2680 

2900 

3219 

136 

3539 

3858 

4177 

4496 

4814 

6133 

6461 

6769 

6086 

6403 

137 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9664 

138 

9879 

.194 

.608 

.822 

1136 

1450 

1763 

2076 

2389 

2702 

139 

143015 

3327 

3630 

3951 

4263 

4574 

4885 

6196 

6607 

5818 

140 

6128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603 

8911 

141 

9219 

9527 

9835 

.142 

.449 

.756 

1063 

1370 

1676 

1982 

142 

162288 

2694 

2900 

3205 

8510 

3815 

4120 

4424 

4728 

6032 

143 

5336 

5640 

5943 

6246 

6649 

6852 

7164 

7457 

7759 

8061 

144 

8362 

8664 

8966 

9266 

9567 

9868 

.168 

.469 

.769 

1068 

146 

161368 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3768 

4056 

146 

4353 

4650 

4947 

6244 

6541 

5838 

6134 

6430 

6726 

7022 

147 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

148 

170262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

149 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

6512 

6802 

4              LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

160 

176091 

6381 

6670 

6959 

7248 

7536 

7825 

8113 

8401 

KIJH9 

151 

8977 

9264 

9552 

9839 

.126 

.413 

.699 

.985 

1272 

1568 

152 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

153 

4691 

4975 

5259 

5542 

5825 

.6108 

6391 

tK>74 

6956 

7-239 

154 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

.  .51 

281 

155 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

156 

3125 

3403 

3681 

3969 

4237 

4514 

4792 

5069 

5346 

6(*3 

157 

5899 

6176 

6453 

6729 

7005 

7281 

-,656 

7832 

8107 

8382 

158 

8657 

8932 

9206 

9481 

9755 

..29 

.303 

.577 

.850 

1124 

159 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3577 

3848 

273 

160 

4120 

4391 

4663 

4934 

5204 

6475 

5746 

6016 

6286 

6656 

161 

6826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

162 

9515 

9783 

..51 

.319 

.586 

.853 

1121 

1388 

1654 

1921 

163 

212188 

2454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

165 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9586 

9846 

166 

220108 

0370 

Ou31 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

167 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

6051 

168 

5309 

5568 

5c26 

6084 

6342 

6000 

6858 

7115 

7372 

7630 

169 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

.193 

257 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

171 

2996 

3250 

3604 

3757 

4011 

4264 

4517 

4770 

5023 

6276 

172 

5528 

5781 

6033 

6286 

6637 

6789 

7041 

7292 

7544 

7796 

173 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

..60 

.300 

174 

240549 

0799 

1048 

1297 

1646 

1795 

2044 

2293 

2641 

2790 

249 

175 

3038 

3286 

3534 

3782 

4030 

4277 

4525 

4772 

6019 

5266 

176 

5513 

5759 

6006 

6252 

6499 

6746 

6991 

7237 

7482 

7728 

177 

7973 

8219 

8464 

8709 

8964 

9198 

9443 

9687 

9932 

.176 

178 

250420 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

2610 

179 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

6031 

242 

* 

180 

5273 

5514 

6755 

6996 

6237 

6477 

6718 

6968 

7198 

7439 

181 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

182 

260071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

183 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

184 

4818 

5054 

5290 

6525 

6761 

6996 

6232 

6467 

6702 

6937 

235 

185 

7172 

7406 

7641 

7876 

8110 

8344 

8578 

8812 

9046 

9279 

186 

9513 

9746 

9980 

.213 

.446 

.679 

.912 

1144 

1377 

1609 

187 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

188 

4158 

4389 

4620 

4850 

6081 

5311 

6542 

6772 

6002 

6232 

189 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8526 

229 

190 

8754 

8982 

9211 

9439 

9667 

9896 

.123 

.351 

.578 

.806 

191 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3076 

192 

3301 

3527 

3763 

3979 

4205 

4431 

4666 

*4882 

5107 

6332 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7678 

194 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

224 

195 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1691 

1813 

2034 

196 

2258 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

197 

4466 

4687 

4907 

6127 

6347 

5567 

5787 

6007 

6226 

6446 

198 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8636 

199 

8853 

9071 

9289 

9607 

9725 

9943 

.161 

.378 

.695 

.813 

OF  LUMBERS           5 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

201 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

202 

5351 

6566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

203 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

204 

9630 

9843 

..66 

.268 

.481 

.693 

.906 

1118 

1330 

1542 

212 

205 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

208 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

209 

320146 

0354 

0662 

0769 

0977 

1184 

1391 

1598 

1805 

2012 

207 

210 

2219 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

211 

4282 

4488 

4694 

4899 

5105 

5310 

651(> 

57-21 

6926 

6131 

212 

6336 

6541 

6745 

6950 

7155 

7369 

7563 

7767 

7972 

8176 

213 

8380 

8583 

8787 

8991 

9194 

9398 

9501 

9805 

...8 

.211 

214 

330414 

0617 

0819 

1022 

1225 

OAO 

1427 

1630 

1832 

2034 

2236 

215 

2438 

2640 

2842 

3044 

£j\Ja 

3246 

3447 

3649 

3850 

4051 

4253 

216 

4454 

4665 

4856 

6057 

5257 

6458 

5658 

5859 

6059 

6260 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

218 

8456 

8656 

8855 

9054 

9253 

9451 

9660 

9849 

.  .47 

.246 

219 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

198 

220 

2423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

221 

4392 

4589 

4786 

4981 

5178 

5374 

6570 

5766 

5962 

6167 

222 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

223 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

.  .54 

224 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

193 

225 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

226 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

227 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

228 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9G4G 

229 

9835 

..25 

.215 

.404 

.593 

.783 

.972 

1161 

1350 

1639 

190 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

T424 

231 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113  5301 

232 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

.  983  7K)9 

233 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845  9030 

234 

9216 

9401 

9587 

9772 

9958 

.143 

.328 

.513 

.698  ;  .883 

185 

235 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

23GO 

2544  '2,28 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382  4565 

237 

4748 

4932 

5115 

6298 

6481 

5664 

5846 

6029 

6212  6394 

238 

6577 

6769 

6942 

7124 

7306 

7488 

7670 

7862 

8034  8216 

239 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849  ,  .  .30 

182 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656  1837 

241 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3466  3636 

242 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5249  5428 

243 

5606 

6785 

6964 

6142 

6321 

6499 

6677 

6856 

V034  7212 

244 

7390 

7668 

7746 

7923 

8101 

8279 

8456 

8634 

8811  8989 

178 

j 

245 

9166 

9343 

9520 

9698 

9876 

..51 

.228 

.405 

.582  .759 

246 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345  2521 

247 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101  4277 

248 

4452 

4627 

4802 

4977 

5162 

5326 

6501 

6676 

5850  6025 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7692  i  7766 

6              LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

250 

397940 

8114 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

9601 

251 

9674 

9847 

..20 

.192 

.365 

.638 

.711 

.883 

1056 

1228 

252 

401401 

1673 

1745 

1917 

2089 

2261 

2433 

ii605 

2777 

2949 

253 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

46(53 

254 

4834 

5005 

5176 

5346 

6617 

6688 

5868 

1)029 

6199 

6370 

171 

255 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

256 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

257 

9933 

.102 

.271 

.440 

.609 

.777 

.946 

1114 

1283 

1451 

258 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

259 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

260 

4973 

5140 

6307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

261 

6641 

6807 

6973 

7139 

',306 

7472 

7638 

7804 

7970 

8135 

262 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

263 

9956 

.121 

.286 

.451 

.616 

.781 

.945 

1110 

1276 

1439 

264 

421604 

1788 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

265 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4655 

4718 

266 

4882 

6045 

6208 

6871 

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6697 

6860 

6023 

6186 

6349 

267 

6511 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

268 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

269 

9752 

9914 

..76 

.236 

.398 

.569 

.720 

.881 

1042 

1203 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

271 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

42,49 

4409 

272 

4569 

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4888 

5048 

6207 

6367 

6526 

6685 

5844 

6004 

273 

6163 

6322 

6481 

6640 

6800 

6957 

7116 

7275 

7433 

7592 

274 

7761 

7909 

8067 

8226 

8384 

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8701 

8869 

9017 

9176 

158 

275 

9333 

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9648 

9806 

9964 

.122 

.279 

.437 

.694 

.752 

276 

440909 

1066 

1224 

1381 

1638 

1695 

1852 

2009 

2166 

.323 

277 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

278 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

6137 

6293 

6449 

279 

5604 

6760 

5915 

6071 

6226 

6882 

6637 

6692 

6848 

7003 

280 

7158 

7313 

7468 

7623 

7778 

7983 

8088 

8242 

8397 

8553 

281 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

..95 

282 

450249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1G33 

283 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3166 

284 

3318 

3471 

3624 

3777 

3930 

4082 

4285 

4387 

4640 

4692 

2S5 

4845 

4997 

5150 

5302 

5454 

5606 

5758 

5910 

6062 

6214 

286 

6366 

6618 

6670 

6821 

6973 

7125 

7276 

7428 

7579 

7731 

287 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8040 

9091 

9242 

288 

9392 

9543 

9694 

9846 

9995 

.146 

.296 

.417 

.597 

.748 

289 

460898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

291 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

292 

5383 

5532  5680 

6829 

6977 

6126 

6274 

6423 

6571 

6719 

293 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

294 

8347 

8495 

8643 

8790 

8938 

9086 

9283 

9380 

9627 

9675 

147 

295 

9822 

9969 

.116 

.263 

.410 

.567 

.704 

.861 

.998 

1146 

296 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

C464 

2(JlO 

297 

2756 

•2903 

3049 

3196 

3341 

8487 

8633 

3779 

3926 

4071 

298 

4216 

'4362 

4508 

4653 

4799 

4944 

6090 

5235 

5381 

6526 

299 

6671 

6816 

6962 

6107 

6262 

6397 

6642 

6687 

6832 

6976 

OF  NUMBERS.              7 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

300 

477121 

7266 

7411 

7555 

7700 

7844 

7989 

8133 

8278 

8422 

301 

8666 

8711 

8855 

8999 

9143 

9287 

9481 

9575 

9719 

9863 

302 

480007 

0161 

0294 

0438 

0682 

0725 

0869 

1012 

1166 

1299 

303 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2446 

2688 

2731 

304 

2874 

3016 

3159 

3302 

3446 

3687 

3730 

3872 

4015 

4167 

142 

305 

4300 

4442 

4585 

4727 

4869 

6011 

6163 

6295 

5437 

6679 

306 

6721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

6997 

307 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

308 

8551 

8692 

8833 

8974 

9114 

9256 

9396 

9537 

9667 

9818 

309 

9959 

..99 

.239 

.380 

.620 

.661 

.801 

.941 

1081 

1222 

310 

491362 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

311 

2760 

2900 

3040 

3179 

3319 

3468 

3597 

3737 

3876 

4015 

312 

4165 

4294 

4433 

4572 

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4850 

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6128 

6267 

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313 

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6683 

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6238 

6376 

6516 

6653 

6791 

314 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

315 

8311 

8448 

8686 

8724 

8863 

8999 

9137 

9275 

9412 

9650 

316 

9687 

9824 

9962 

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.236 

.374 

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.648 

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.922 

317 

601059 

1196 

1333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

318 

2427 

2564 

2700 

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2973 

3109 

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3382 

3518 

3655 

319 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

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1878 

6014 

320 

5150 

6286 

5421 

5657 

5693 

6828 

5964 

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6234 

6370 

321 

6505 

6640 

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7181 

7316 

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7586 

7721 

322 

7856 

7991 

8126 

8260 

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8630 

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8934 

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323 

9203 

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9471 

9606 

9740 

9874 

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324 

610546 

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0813 

0947 

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1349 

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1/60 

134 

326 

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2017 

2161 

2284 

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2551 

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2818 

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326 

3218 

3351 

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3617 

3750 

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4149 

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327 

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5609 

5741 

328 

5874 

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6403 

6636 

6668 

6800 

6932 

7054 

329 

7196 

7328 

7460 

7692 

7724 

7855 

7987 

8119 

8251 

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330 

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8646 

8777 

8909 

9040 

9171 

9303 

9434 

9566 

9697 

331 

9828 

9959 

..90 

.221 

.353 

.484 

.615 

.746 

.876 

1007 

332 

521138 

1269 

1400 

1630 

1661 

1792 

1922 

2053 

2183 

2314 

333 

2444 

2575 

2706 

2835 

S966 

3096 

3226 

3356 

3486 

3616 

334 

3746 

3876 

4006 

4136 

4266 

4396 

4526 

4666 

4785 

4915 

335 

5045 

5174 

5304 

5434 

5563 

6693 

6822 

6951 

6081 

6210 

336 

6339 

6469 

6698 

6727 

6856 

6985 

7114 

7243 

7372 

7501 

337 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

8531 

8660 

8788 

338 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9816 

9943 

..72 

339 

630200 

0328 

0456 

0584 

0712 

0840 

0968 

1096 

1223 

Io61 

340 

1479 

1607 

1734 

1862 

1960 

2117 

2245 

2372 

2500 

2027 

341 

2754 

2882 

3009 

3136 

3264 

3391 

3618 

3645 

3772 

3899 

342 

4026 

4163 

4280 

4407 

4534 

4651 

4787 

4914 

6041 

5167 

343 

5294 

6421 

6647 

5674 

5800 

6927 

6053 

6180 

6306 

6432 

344 

6558 

6686 

6811 

6937 

7060 

71b9 

7316 

7441 

/567 

7693 

129 

345 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

346 

9076 

9202 

9327 

9452 

9678 

9703 

9829 

9954 

..79 

.204 

347 

640329 

0455 

0580 

0705 

0830 

0955 

1080 

1205 

1330 

1454 

348 

1579 

1704 

1829 

1963 

2078 

2203 

2327 

2462 

2676 

2/01 

349 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3oW> 

3»*0 

3^44 

8              LOGARITHMS 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

350 

544068 

4192 

4316 

4440 

4564 

4688 

4812 

4936 

6060 

5183 

351 

5307 

5431 

5555 

6678 

5805 

5925 

6049 

6172 

6296 

6419 

352 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

363 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8636 

8758 

8881 

354 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

.196 

122 

355 

550228 

0351 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

350 

1450 

1672 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

357 

2o(S8 

2790 

2911 

3033 

3165 

3276 

3393 

3519  !  3G40 

3762 

358 

3883 

4004 

4126 

4247 

4368' 

4489 

4610 

4731 

4852 

4973 

359 

6094 

5215 

5346 

6467 

6578 

5699 

6820 

5940 

6061 

6182 

360 

6303 

G423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7G87 

301 

7507 

7W27 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

3G2 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

8o3 

9907 

26 

.146 

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.385 

.504 

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.743 

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.982 

364 

561101 

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1340 

1459 

1678 

1698 

1817 

1936 

2065 

2173 

365 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

366 

8481 

3600 

3718 

3837 

3965 

4074 

4192 

4311 

4429 

4648 

367 

4666 

4784 

4903 

6021 

5139 

5257 

6376 

5494 

6612 

5730 

368 

6848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

370 

8202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

9257 

371 

9374 

9491 

9608 

9725 

9882 

9959 

..76 

.193 

.309 

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372 

570543 

0660 

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0893 

1010 

1126 

1243 

1359 

1476  |  1592 

373 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

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2639 

2755 

374 

2872 

2988 

3104 

3220 

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3568 

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376 

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6226 

377 

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7377 

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8410 

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8639 

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8868 

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9555 

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380 

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.126 

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.356 

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.811 

381 

580925 

1039 

1153 

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1381 

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195* 

382 

2063 

2177 

2291 

2404 

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2631 

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2972 

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383 

3199 

3312 

3426 

3539 

3652 

3766 

3879 

3992 

4105 

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4331 

4444 

4567 

4670 

4783 

4896 

6009 

6122 

5235 

5348 

385 

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5674 

6686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

386 

6587 

6700 

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6926 

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7262 

7374 

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7699 

387 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

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8832 

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9056 

9167 

9279 

9391 

9603 

9615 

9726 

9834 

389 

9950 

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.173 

.284 

.396 

.607 

.619 

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.842 

.953 

390 

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1176 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2066 

391 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3176 

392 

3286 

3397 

3508 

3618 

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393 

4393 

4503 

4614 

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5165 

6276 

6386 

394 

6496 

5606 

6717 

6827 

5937 

6047 

6167 

6267 

6377 

6487 

110 

395 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

396 

7696 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

397 

8791 

8900 

9009 

9119 

9228 

i.337 

9446 

.-556 

9666 

L'774 

398 

9883 

9992 

.101 

.210 

.319 

.428 

.637 

.646 

755 

.864 

399 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

OF  NU-MBERS.              9 

N. 

0 

1 

2 

3  |  4 

5 

6 

7 

8 

9 

400 

602060 

2169 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3036 

401 

3144 

3253 

3361 

3469 

3573 

3686 

3794 

3902 

4010 

4118 

402 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

5089 

5197 

403 

5305 

5413 

5521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

404 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

108 

406 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

406 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

407 

9594 

9701 

9808 

9914 

..21 

.128 

.234 

.341 

.447 

.654 

408 

610660 

0767 

0873 

0979. 

1086 

1192 

1298 

1405 

1511 

1617 

409 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

410 

2784 

2890 

2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

411 

3842 

3947 

4053 

4159 

4264 

4370 

4476 

4581 

4686 

4792 

412 

4897 

5003 

5108 

5213 

5319 

5424 

5629 

5634 

6740 

5845 

413 

6950 

6055 

6160 

6266 

6370 

6476 

6581 

6686 

6790 

6896 

414 

7000 

7105 

7210 

7316 

7420 

7625 

7629 

7734 

7839 

7943 

415 

8048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

416 

9293 

9198 

9302 

9406 

9511 

9615 

9719 

9824 

9928 

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417 

620136 

0140 

0344 

0448 

0552 

0656 

0760 

0864 

0968 

1072 

418 

1176 

1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

419 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

422 

6312 

6415 

6518 

5621 

6724 

5827 

5929 

6032 

6135 

6238 

423 

6340 

6443 

6546 

6648 

6761 

6853 

6956 

7058 

7161 

7263 

424 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

103 

426 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

426 

9410 

9512 

9613 

9715 

9817 

9919 

..21 

.123 

.224 

.326 

427 

630428 

0530 

0631 

0733 

0835 

0936 

1038 

1139 

1241 

1342 

428 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

430 

3468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

431 

4477 

4578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

6383 

432 

6484 

6584 

5685 

6785 

5886 

6986 

6087 

6187 

6287 

6388 

433 

6488 

6688 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

7390 

434 

7490 

7690 

7690 

7790 

78*90 

7990 

8090 

8190 

8290 

8389 

436 

8489 

8589 

8689 

8789 

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9088 

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9287 

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9486 

9586 

9686 

9785 

9885 

9984 

..84 

.183 

.283 

.382 

437 

640481 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

438 

1474 

1573 

1672 

1771 

1871 

1970 

20179 

2168 

2267 

2366 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

440 

3453 

3551 

3650 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

441 

4439 

4537 

4636 

4734 

4832 

4931 

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5127 

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53L»4 

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6422 

5521 

.6619 

6717 

5815 

5913 

6011 

6110 

0208 

(iSUti 

443 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7<M 

7187 

7285 

444 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

81(-:5 

8262 

93 

445 

8360 

8458 

8555 

8653 

8750 

8848 

8945 

90  43 

9140 

9237 

446 

9385 

9432 

9530 

9627 

9724 

9821 

9919 

.  .  Iti 

.113 

.210 

447 

650308 

0405 

0502 

0599 

0696 

0793 

0890 

0987 

1084 

1181 

448 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

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•2053 

2150 

449 

2246 

2343 

2440 

2530 

2633 

2730 

2826 

2923 

3019 

3116 

10             LOGARITHMS 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

460 

653213 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

451 

4177 

4273 

4369 

4466 

4562 

4668 

4764 

4850 

4946 

6042 

462 

6138 

5235 

6331 

5427 

5526 

5019 

5715 

5810 

6906 

6002 

463 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

454 

7056 

7152 

7247 

7343 

7438 

7534 

7(529 

7725 

7820 

7916 

96 

455 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

456 

8966 

9060 

9155 

9250 

9340 

:>441 

'..530 

9031 

9726 

^821 

457 

9916 

..11 

.106 

.201 

.290 

.391 

.486 

.581 

.676 

.771 

458 

660865 

0960 

1055 

1150 

1245 

1339 

1434 

1629 

1623 

1718 

469 

1813 

1907 

2002 

20JO 

2191 

2280 

2380 

2475 

2669 

2663 

460 

2768 

2852 

2947 

3041 

3135 

3230 

3324 

3418 

3512 

8607 

461 

3701 

3796 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4648 

462 

4642 

4736 

4830 

4924 

6018 

5112 

5206 

6299 

6393 

6487 

463 

5581 

6675 

5769 

6862 

5956 

6050 

6143 

6237 

6331 

6424 

464 

6618 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

465 

7453 

7646 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

466 

8386 

8479 

8572 

8666 

8759 

8852 

8945 

9038 

9131 

9324 

467 

9317 

9410 

9503 

9596 

9689 

9782 

9876 

9967 

..60 

.153 

468 

670241 

0339 

0431 

0524 

0617 

0710 

0802 

0895 

0988 

1080 

469 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2006 

470 

2098 

2190 

2283 

2375 

2467 

2560 

2662 

2744 

2836 

2929 

471 

3021 

3113 

3205 

3297 

3390 

3482 

3574 

3666 

3768 

3850 

472 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

473 

4861 

4963 

6045 

5137 

6228 

5320 

5412 

6603 

5595 

6687 

474 

5778 

6870 

6962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

91 

476 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7616 

476 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

477 

8518 

8609 

8700 

8791 

8882 

8972 

9064 

9165 

9246 

9337 

478 

9428 

9619 

9610 

9700 

9791 

9882 

9973 

..63 

.154 

.246 

479 

680336 

0426 

0517 

0607 

0698 

0789 

0879 

0970 

1060 

1151 

480 

1241 

1332 

1422 

1513  ' 

1603 

1693 

1784 

1874 

1964 

2055 

481 

2146 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2967 

482 

3047 

3137 

3227 

3317 

3407 

3497 

3687 

3677 

3767 

3867 

483 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

484 

4854 

4935 

5026 

5114 

5204 

6294 

6383 

6473 

6663 

5652 

485 

5742 

6831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

486 

6636 

6726 

6816 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

487 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

488 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

489 

9309 

9398 

9486 

9576 

9664 

9753 

9841 

9930 

..19 

.107 

490 

690196 

0285 

0373 

0962 

0550 

0639 

0728 

0816 

0905 

0993 

491 

1081 

1170 

1268 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

492, 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

493 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

494 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

88 

495 

4605 

4693 

4781 

4868 

4956 

6044 

5131 

5210 

5307 

6394 

496 

5482 

6569 

6657 

6744 

5832 

6919 

6007 

6094 

6182 

6269 

497 

6356 

5444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

498 

7229 

7317 

7404 

7491 

7678 

7666 

7752 

7839 

7926 

8014 

499 

8101 

8188 

8275 

8362 

8449 

8535 

8632 

8709 

8796 

8883 

OF  NUMBERS.             11 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

500 

698970 

9057 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

501 

9838 

9924 

..11 

..98 

.184 

.271 

.358 

.444 

.531 

.617 

502 

700704 

0790 

0877 

0963 

1050 

1136 

1222 

1309 

1395 

1482 

503 

1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

504 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

86 

505 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3895 

3979 

4065 

606 

4151 

4236 

4322 

4408 

4494 

4679 

4665 

4751 

4837 

4922 

507 

5008 

5094 

6179 

5265 

6350 

6436 

5622 

5607 

5693 

5778 

508 

6864 

5949 

6035 

6120 

6206 

6291 

6376 

6462' 

6547 

6632 

509 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7316 

7400 

7485 

510 

7570 

7655 

7740 

7826 

7910 

7996 

8081 

8166 

8251 

8336 

611 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

512 

9270 

9365 

9440 

9524 

9609 

9694 

9i  79 

9863 

9948 

..33 

513 

710117 

0202 

0287 

0371 

0456 

0540 

0625 

0710 

0794 

0879 

514 

0963 

1048 

1132 

1217 

1301 

1385 

1470 

1654 

1639 

1723 

515 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2566 

616 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3326 

340  r 

517 

3491 

3575 

3659 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

618 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916 

5000 

5084 

619 

6167 

5251 

5335 

5418 

6502 

6586 

5669 

6753 

6836 

5920 

520 

6003 

6087 

6170 

6254 

6337 

6421 

6504 

6588 

6671 

6754 

621 

6838 

6921 

7004 

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7171 

7254 

7338 

7421 

7504 

7687 

622 

7671 

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7837 

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8003 

8086 

8169 

8253 

8336 

8419 

623 

8502 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

9166 

9248 

624 

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9414 

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9680 

9663 

CO 

9745 

9828 

9911 

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625 

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0325 

0407 

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0490 

0573 

0655 

0738 

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0903 

626 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

527 

1811 

1893 

.975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

528 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

529 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

530 

4276 

4358 

4440 

4622 

4604 

4685 

4767 

4849 

4931 

6013 

531 

6095 

5176 

6258 

6340 

6422 

5503 

6585 

5667 

5748 

6830 

532 

6912 

6993 

6075 

6156 

6238 

6320 

6401 

6483 

6664 

6646 

633 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

534 

7541 

7623 

7704 

7786 

7866 

7948 

8029 

8110 

8191 

8273 

535 

8354 

8435 

8516 

8597 

8678 

8769 

8841 

8922 

9003 

9084 

536 

9165 

9246 

9327 

9403 

9489 

9570 

9651 

9732 

9813 

9893 

537 

9974 

..55 

.136 

.217 

.298 

.378 

.459 

.440 

.621 

.702 

638 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

539 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

640 

2394 

2474 

2565 

2635 

2715 

2796 

2876 

2956 

3037 

3117 

641 

3197 

3278 

3358 

3438 

3518 

3598 

3679 

3769 

3839 

3919 

542 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

543 

4800 

4S80 

4960 

5040 

6120 

5200 

5279 

5359 

5439 

6619 

644 

5599 

5679 

6769 

5838 

5918 

6998 

6078 

6157 

6237 

6317 

80 

645 

6397 

6476 

6556 

6636 

6715 

6796 

6874 

6954 

7034 

7113 

546 

7193 

7272 

7362 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

647 

7987 

8067 

8146 

8225 

8305 

8384 

8463 

8543 

8622 

8701 

548 

8781 

8860 

8939 

9018 

9097 

9177 

9266 

9336 

9414 

9493 

649 

9672 

9651 

9731 

9810 

9889 

9968 

..47 

.126 

.205 

.284 

17 


12             LOGARITHMS 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

650 

740363 

0442 

0521 

0560 

0678 

0767 

0836 

0915 

0994 

1073 

651 

1152 

1230 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

186U 

562 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2(346 

663 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3363 

3431 

664 

3510 

3558 

3667 

3745 

3823 

3902 

3980 

4058 

4136 

4216 

79 

655 

4293 

,4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

666 

6075 

6153 

5231 

6309 

5387 

6466 

6543 

5621 

5699 

6777 

557 

6855 

5933 

6011 

6089 

6167 

6246 

6323 

6401 

6479 

6556 

658 

6634 

6712 

6790 

6868 

6946 

7023 

7101 

7179 

7256 

7334 

659 

7412 

7489 

7667 

7645 

7722 

7800 

7878 

7966 

8033 

8110 

560 

8188 

8266 

8343 

8421 

8498 

8576 

8653 

8731 

8808 

8885 

661 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

662 

9736 

9814 

9891 

9968 

..46 

.123 

.200 

.277 

.354 

.431 

563 

760508 

0586 

0663 

0740 

0817 

0894 

0971 

1048 

1125 

1202 

664 

1279 

1366 

1433 

1510 

1687 

1664 

1741 

1818 

1895 

1972 

665 

2048 

2125 

2202 

2279 

2356 

2433 

2609 

2586 

2663 

2740 

566 

2816 

2893 

2970 

3047 

3123 

3200 

3277 

3363 

3430 

3506 

667 

3582 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4196 

4272 

668 

4348 

4425 

4501 

4678. 

4654 

4730 

4807 

4883 

4960 

5036 

569 

5112 

6189 

5265 

5341 

5417 

6494 

5570 

5646 

6722 

6799 

570 

6875 

5961 

6027 

6103 

6180 

6266 

6332 

6408 

6484 

6560 

671 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

572 

7396 

7472 

7548 

7624 

7700 

7776 

7861 

7927 

8003 

8079 

673 

8165 

8230 

8306 

8382 

8458 

8633 

8609 

8685 

8761 

8836 

674 

8912 

8988 

9068 

9139 

9214 

9290 

9366 

9441 

9617 

9692 

74 

676 

9668 

9743 

9819 

9894 

9970 

..45 

.121 

.196 

.272 

.347 

676 

760422 

0498 

0573 

0649 

0724 

0799 

0876 

0960 

1026 

1101 

677 

1176 

1251 

1326 

1402 

1477 

1662 

1627 

1702 

1778 

1853 

678 

1938 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

679 

2679 

2764 

2829 

2904 

2978 

3053 

3128 

2203 

3278 

3353 

680 

3428 

3503 

3578 

3653 

3727 

3802 

3877 

3952 

4027 

4101 

681 

4176 

4251 

4326 

4400 

4475 

4560 

4624 

4699 

4774 

4848 

582 

4923 

4998 

6072 

5147 

6221 

5296 

5370 

5445 

6520 

6594 

583 

5669 

5743 

6818 

5892 

6966 

6041 

6115 

6190 

6264 

6338 

684 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

585 

7166 

7230 

7304 

7379 

7453 

7627 

7601 

7675 

7749 

7823 

586 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490 

8664 

687 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

688 

9377 

9451 

9525 

9699 

9673 

9746 

9820 

9894 

9968 

..42 

589 

770115 

0189 

0263 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

590 

0852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

691 

1587 

1661 

1734 

1808 

1881 

1956 

2028 

2102 

2176 

2248 

692 

2322 

2396 

2468 

3542 

2615 

2688 

2762 

2836 

2908 

2981 

593 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

594 

3786 

3860 

3933 

4006 

4079 

4162 

4226 

4298 

4371 

4444 

73 

595 

4617 

4590 

4663 

4736 

4809 

4882 

4955 

6028 

6100 

6173 

596 

6246 

5319 

5392 

6466 

5538 

6610 

6683 

6756 

6829 

6902 

697 

6974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

698 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

599 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

OF  NUMBERS.             13 

N. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

600 

778151 

8224 

8296 

8368 

8441 

8513 

8685 

8658 

8730 

8802 

601 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

602 

9596 

6669 

9741 

9813 

9885 

9957 

..29 

.101 

.173 

.245 

603 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0966 

604 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

72 

605 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

606 

2473 

2544 

2616 

2688 

2769 

2831 

2902 

2974 

3046 

3117 

607 

3189 

3260 

3332 

3403 

3476 

3546 

3618 

3689 

3761 

3832 

608 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

609 

4617 

4689 

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OF  NUMBERS.             15 

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900367 

0422 

0476 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

796 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

797 

1458 

1513 

1567 

1622 

1676 

1736 

1786 

1840 

ltt>4 

1948 

798 

2003 

2057 

2112 

2166 

2221 

2276 

2329 

2384 

2438 

2492 

799 

2647 

2601 

2666 

2710 

2764 

2818 

2873 

2927 

2081 

3036 

OF  NUMBERS.              17 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8  |   9 

800 

903090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

801 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4563 

4607 

4661 

803 

4716 

4770 

4824 

4878 

4932 

4986 

6040 

5094 

5148 

6202 

804 

5256 

5310 

6364 

5418 

6472 

5526 

6580 

5634 

5688 

6742 

54 

805 

5796 

5850 

6904 

6958 

6012 

6066 

6119 

6173 

6227 

6281 

806 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

807 

6874 

6927 

6981 

7036 

7089 

7143 

7196 

7250 

7304 

7368 

808 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

809 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

810 

8485 

8539 

8592 

8646 

8699 

8763 

8807 

8860 

8914 

8967 

811 

9021 

9074 

9128 

9181 

9236 

9289 

9342 

9396 

9449 

9503 

812 

9556 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

..37 

813 

910091 

0144 

0197 

0251 

0304 

0358 

0411 

0464 

0618 

0571 

814 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

815 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

816 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

817 

2222 

2275 

2323 

2381 

2435 

2488 

2541 

2594 

2645 

2700 

818 

2753 

2806 

2869 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

819 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3665 

3708 

3761 

820 

3814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

821 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

822 

4872 

4925 

4977 

6030 

6083 

5136 

5189 

6241 

6594 

6347 

823 

5400 

5453 

5605 

6558 

5611 

5664 

6716 

5769 

6822 

6875 

824 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

825 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

826 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

827 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

828 

8030 

8083 

8185 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

829 

8655 

8607 

8669 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

830 

9078 

9130 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

831 

9601 

9663 

9706 

9758 

9810 

9862 

9914 

9967 

..19 

..71 

832 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

833 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

834 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1630 

1582 

1634 

835 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

836 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

837 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

838 

3244 

3296 

3348 

3399 

3451 

3503 

3565 

3607 

3658 

3710 

839 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4147 

4228 

840 

4279 

4331 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

841 

4796 

4848 

4899 

4951 

6003 

5054 

5106 

5157 

5209 

5261 

842 

5312 

5364 

5415 

5467 

6618 

5570 

5621 

5673 

5725 

5776 

843 

5828 

5874 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

844 

6342 

6394 

6446 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

52 

845 

6857 

6908 

6959 

7011 

7062 

7114 

71G5 

7216 

7268 

7319 

846 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7783 

7832 

847 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

848 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8803 

8857 

849 

8908 

8959 

9010 

9061 

9112 

9163 

9216 

9266 

9317 

9368 

18             LOGARITHMS 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

850 

929419 

9473  9521 

9672 

9623 

9674 

9726 

9776 

9827 

9879 

851 

9930 

9981 

..32 

..83 

.134 

.186 

.236 

.287 

.338 

.389 

852 

930440 

0491 

0542 

0592 

0643 

0694 

0746 

0796 

0847 

0898 

853 

0949 

1000 

1051 

1102 

1163 

1204 

1254 

1305 

1356 

1407 

854 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1866 

1916 

51 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

856 

2474 

2524 

2675 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

857 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3336 

3386 

3437 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

859 

3993 

4044 

4094 

4145 

4195 

4246 

4269 

4347 

4397 

4448 

860 

4498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

861 

5003 

6054 

5104 

6164 

5205 

5255 

5306 

5356 

5406 

6457 

862 

5507 

5558 

5608 

5668 

5709 

5769 

5809 

6860 

6910 

6960 

863 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

864 

6514 

6564 

6614 

6665 

6715 

6766 

6816 

6866 

6916 

6966 

865 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

866 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

868 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8919 

8970 

869 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

870 

9519 

9569 

9616 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

871 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

872 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0916 

0964 

873 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

874 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

875 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2365 

2405 

2465 

876 

2504 

2554 

2603 

2653 

2702 

2762 

2801 

2851 

2901 

2950 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

878 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

880 

4483 

4532 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

881 

4976 

5025 

6074 

5124 

6173 

5222 

6272 

6321 

6370 

6419 

882 

5469 

5518 

6667 

6616 

6665 

5715 

5764 

5813 

6862 

5912 

883 

5961 

6010 

6059 

6108 

6157 

6207 

6266 

6305 

6354 

6403 

884 

6452 

6501 

6661 

6600 

6649 

6698 

6747 

6796 

6846 

6894 

885 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

886 

7434 

7483 

7632 

7681 

7630 

7679 

7728 

7777 

7826 

7875 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8365 

888 

8413 

8462 

8611 

8560 

8609 

8657 

8706 

8/65 

8804 

8853 

889 

8902 

8961 

8999 

9048 

9097 

9146 

9196 

9244 

9292 

9341 

890 

9390 

9439 

9488 

9536 

9685 

9634 

9683 

9731 

9780 

9829 

891 

9878 

9926 

9975 

..24 

..73 

.121 

.1/0 

.219 

.267 

.316 

892 

950365 

0414 

0462 

0511 

0660 

0608 

0667 

0706 

0754 

0803 

893 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

894 

1338 

1386 

1436 

1483 

1632 

1580 

1629 

1677 

1726 

17/5 

48 

895 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

896 

2308 

2366 

2406 

2463 

2602 

2550 

2599 

2647 

5696 

2744 

897 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

898 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3616 

3663 

3711 

899 

3760 

3808 

3856 

3905 

3963 

4001 

4019 

4098 

4146 

4194 

OF  NUMBERS.             19 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628 

4677 

901 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

902 

6207 

5255 

5303 

6361 

5399 

6447 

5495 

5543 

6592 

6640 

903 

6688 

5736 

6784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

904 

6168 

6216 

6266 

6313 

6361 

6409 

6457 

6505 

6653 

6601 

48 

905 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

906 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7612 

7559 

907 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

908 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

909 

8564 

8612 

8659 

8707 

8755 

8803 

8860 

8898 

8946 

8994 

910 

9041 

9089 

9137 

9186 

9232- 

9280 

9328 

9375 

9423 

9471 

911 

9518 

9566 

9614 

9661 

9709 

9767 

9804 

9852 

9900 

9947 

912 

9995 

..42 

..90 

.138 

.185 

.233 

.280 

.328 

.376 

.423 

913 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

914 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

915 

1421 

1469 

1516 

1663 

1611 

1658 

1706 

1753 

1801 

1848 

916 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2276 

2322 

917 

2369 

2417 

2464 

2511 

2569 

2606 

2653 

2701 

2748 

2796 

918 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

919 

3316 

3363 

3410 

3467 

3504 

3552 

3599 

3646 

3693 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

922 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

6108 

5165 

923 

5202 

5249 

5296 

5343 

6390 

5437 

5484 

5531 

5578 

6625 

924 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

926 

6611 

6658 

6705 

6752 

6799 

6846 

6892 

6939 

6986 

7033 

927 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

928 

7548 

7595 

7642 

7688 

7736 

7782 

7829 

7875 

7922 

7969 

929 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

930 

8483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

931 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

932 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

933 

9882 

9928 

9975 

..21 

..68 

.114 

.161 

.207 

.254 

.300 

934 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0766 

935 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

936 

1276 

1322 

1369 

1416 

1461 

1508 

1554 

1601 

1647 

1693 

937 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2167 

938 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

939 

2666 

2712 

2768 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

3313 

3359 

3405 

3451 

3497 

3543 

941 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

943 

4612 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

944 

4972 

6018 

6064 

6110 

6156 

6202 

5248 

5294 

5340 

6386 

46 

945 

5432 

5478 

5524 

6570 

5616 

5662 

5707 

5753 

5799 

6845 

946 

6891 

6937 

6983 

6029 

6076 

6121 

6167 

6212 

6258 

6304 

947 

6350 

6396 

6442 

6488 

6533 

6579 

6925 

6671 

6717 

6763 

948 

6803 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7176 

7220 

949 

7266 

7312 

7368 

7403 

7449 

7496 

7641 

7586 

7632 

7678 

20              LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

950 

977724 

7769 

7816 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

951 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

952 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8958 

9002 

9047 

953 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

954 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

955 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

956 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0321 

0867 

957 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

958 

1366 

1411 

1456 

1601 

1547 

1592 

1637 

1683 

1728 

1773 

959 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

961 

2723 

2769 

2814 

2853 

2904 

2949 

2994 

3040 

3085 

3130 

962 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

963 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

984 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

965 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

966 

4977 

5022 

5087 

5112 

6157 

5202 

5247 

5292 

6337 

5382 

967 

5426 

5471 

5516 

5561 

5606 

5651 

5699 

5741 

5786 

5830 

968 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

6906 

6951 

6996 

7040 

7035 

7130 

7175 

971 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7632 

7577 

7622 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8088 

973 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8426 

8470 

8514 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

891G 

8960 

975 

9005 

9049 

9093 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

976 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

977 

9895 

9939 

9983 

..28 

..72 

.117 

.161 

.206 

.250 

.294 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

981 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2087 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2405 

2509 

983 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4626 

46(59 

4713 

988 

4757 

4801 

4845 

4886 

4933 

4977 

5021 

5065 

5108 

6152 

989 

5196 

5240 

5284 

5328 

5372 

6416 

5460 

6504 

5547 

5691 

990 

5635 

5679 

5723 

6767 

6811 

5854 

6893 

6942 

5986 

6030 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

992 

6512 

6555 

6699 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

994 

7386 

7430 

7474 

7617 

7661 

7605 

7648 

7692 

7736 

7779 

44 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

997 

8695 

8739 

8792 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

998 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9436 

9479 

9622 

999 

9565 

9609 

9662 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

TABLE  II.    Log.  Sines  and  Tangents.  (0°)  Natural  Sines.          21 

' 

Sine. 

D.10" 

Cosine.  ' 

D.10" 

Tang. 

D.10" 

Coiang. 

N.sine 

N.  cos. 

o 

Minns  inf. 

10.000000 

Minus  inf. 

Infinite. 

oooou 

100000 

60 

1 

6.463726 

000000 

6.463726 

13.536274 

00029 

100000 

59 

2 

764766 

000000 

764756 

235244 

00058 

100000 

58 

3 

940847 

000000 

940847 

059153 

00087 

100000 

57 

4 

7.065786 

000000 

7.066786 

12.934214 

00116 

100000 

56 

5 

162696 

000000 

162696 

837304 

00145 

100000 

65 

6 

241877 

9.999999 

241878 

758122 

00175 

100000 

54 

7 

308824 

999999 

308825 

691175 

00204 

100000 

63 

8 

366816 

999999 

366817 

633183 

00233 

100000 

62 

9 

417968 

999999 

417970 

682030 

00262 

100000 

51 

10 

463725 

999998 

463727 

636273 

00291 

100000 

50 

11 

7.505118 

9.999998 

7.605120 

12.494880 

00320 

99999 

49 

12 

542906 

999997 

542909 

457091 

00349 

99999 

48 

13 

577668 

999997 

677672 

422328 

00378 

99999 

47 

14 

609853 

999996 

609857 

390143 

0040- 

99999 

46 

15 

639816 

999996 

639820 

360180 

00436 

99999 

45 

16 

667846 

999995 

667849 

332151 

00465 

99999 

44 

17 

694173 

999995 

694179 

305821 

00495 

99999 

43 

18 

718997 

999994 

719003 

280997 

00524 

99999 

42 

19 

742477 

999993 

742484 

257516 

00553 

99998 

41 

20 

764754 

999993 

764761 

235239 

00582 

99998 

40 

21 

7.785943 

9.999992 

7.785951 

12.214049 

00611 

99998 

39 

22 

806146 

999991 

806156 

193845 

00640 

99998 

38 

23 

825451 

999990 

825460 

174540 

00869 

99998 

37 

24 

843934 

999989 

843944 

156056 

00698 

99998 

36 

25 

861663 

999988 

861674 

138326 

00727 

9999? 

35 

26 

878695 

999988 

878708 

121292  i  00756 

99997 

34 

27 

895085 

999987 

895099 

104901 

00785 

99997 

33 

28 

910879 

999986 

910894 

089106 

00814 

99997 

32 

29 

926119 

999985 

926134 

073866 

00844 

99996 

31 

30 

940842 

999983 

940858 

059142 

00873 

99996 

30 

31 

32 

7.955082 
968870 

2298 

9.999982 
999981 

0.2 

7.955100 
968889 

2298 

12.044900 
031111 

00902 
00931 

99996 
99996 

29 

•28 

33 

982233 

2227 

999980 

0.2 

982263 

2227 

017747 

00960 

99995 

27 

34 

995198 

2161 

999979 

0.2 

995219 

2161 

004781 

00989 

99995 

26 

35 

8.007787 

2098 

999977 

0-2 

8.007809 

2098 

11  .992191 

01018 

99995 

25 

36 

020021 

2039 

999976 

0-2 

020045 

2039 

979955 

01047 

99995 

24 

37 

031919 

1983 

999975 

0-2 

031945 

1983 

968055 

01076 

99994 

'23 

38 
39 
40 
41 

043501 
054781 
065776 
8.076500 

1930 
1880 
1832 

1787 

999973 
999972 
999971 
9.999969 

0-2 
0-2 
0-2 
0'2 

043527 
054809 
085806 
8.076531 

1930 
1880 
1833 

1787 

956473  01105 
945191  01134 
9341941  01164 
11.923469!  01193 

99994 
99994 
99993 
99993 

'22 
21 
•20 
19 

42 

086965 

1744 

999968 

0'2 

086997 

1744 

913003  ' 

01222 

99993 

18 

43 

44 

097183 
107167 

1703 
1664 

999966 
999964 

0'2 

097217 
107202 

1703 
1664 

902783 
892797 

01251 
01280 

99992 
99992 

17 
16 

45 
46 

116926 
126471 

1626 
1591 

999963 
999961 

Q'3 

116963 
126510 

1627 
1591 

883037  1  01309 
873490  01338 

99991 
99991 

16 
14 

47 

135810 

1557 

1  KO/i 

999959 

0.3 

135851 

1557 

864149  01367 

99991 

13 

48 
49 

144953 
153907 

Ioz4 
1492 

999958 
999956 

0^3 

144996 
153952 

1624 
1493 

855004  01396 
846048  01426 

99990 
99990 

12 
11 

50 

162681 

1462 

999954 

0.3 

162727 

1463 

837273  !  !  01454 

99989 

10 

51 

8.171280 

1433 

9.999952 

0.3 

8.171328 

1434 

11.  828672!  101483 

99989 

9 

52 
53 

179713 

187985 

1405 
1379 

999960 
999948 

0.3 
0.3 

179763 
188036 

1406 
1379 

820237  [01513 
811964|;01542 

99989 
99988 

8 

7 

54 

196102 

1353 

999946 

0.3 

196156 

1353 

803844  i  01571 

99988 

6 

55 

204070 

1328 

999944 

0.3 

204126 

1328 

795874;  '01600 

99987 

6 

66 

211895 

1304 

999942 

0.3 

211953 

1304 

788047  '101629 

99987 

4 

67 

58 

219581 
227134 

1281 
1259 

999940 
999938 

0.4 
0.4 

219641 
227195 

1281 
1269 

780359  i  01658 
772805  !!  01687 

99986 
99986 

3 

2 

59 

234557 

1237 

999936 

0.4 

234621 

1238 

765379-01716 

99986 

1 

60 

241855 

1216 

999934 

0.4 

241921 

1217 

768079  jj  01745 

99985 

0 

Cosine. 

Sine. 

Golan?. 

Tan?.    N.  cos. 

\.  sine- 

i 

89  Degrees. 

22          Log.  Sines  and  Tangents.  (1°)  Natural  Sines.   TABLE  II. 
-  . 

Sine. 

D.10" 

Cosine. 

D.10" 

Tang. 

D.10" 

Cotang.  I  IN.  sine. 

N.  cos. 

0 

8.241855 

1  1QR 

9.999934 

OA 

8.241921 

1  1Q7 

11.7580791  01742 

09985 

60 

1 

249033 

iiyo 

1  1  77 

999932 

,*± 
OA 

249102 

L  iy  / 

U77 

750898. 

01774 

)9984|  59 

2 
3 

256094 
263042 

Hi/ 

1158 
1  1  /in 

999929 
999927 

.ft 

0.4 

OA 

256165 
263115 

/  i 
1158 

1  1  AH 

743835 
•  736885 

01803 
01832 

999841  58 
999831  57 

4 

269881 

114U 

999925 

,*l 

269956 

1  14tU 

730044 

01862 

99983  66 

6 
6 

7 

276614 
283243 
289773 

1122 
1105 

1088 

999922 
999920 
999918 

0.4 
0.4 
0.4 

OA 

276691 
283323 
289856 

1122 
1105 
1089 

723309 

716577 
710144 

01891 
0192U 

99982!  55 
99982!  54 
99*81!  63 

8 

296207 

1072 

1  ftcc 

999916 

.4 
04. 

296292 

1073 

703708 

i  01  9  /b 

!>H980i  52 

9 

302546 

-lUOO 

i  n/i  i 

999913 

.  ft 
04. 

302634 

IfUQ 

6973o!J 

0-200, 

399801  51 

10 

308794 

1U41 
1  097 

999910 

•  rr 

04. 

308884 

1  027 

691  1  16 

1  0203( 

999';  9!  50 

11 

8.314954 

1  '  'w  / 

9.999907 

•  -t 

OA 

8.315046 

L\)Z  t 

11  .684J354 

O^t'bc 

:M)79'  49 

12 

321027 

QQQ 

999905 

•  Q 

04 

321122 

999 

678878 

j  02094 

&078|  48 

13 

327016 

yyo 

999902 

•  ^ 

327114 

672886 

J0212S 

99977 

47 

14 

332924 

985 

999899 

0.4 
Oef 

333025 

985 

666975 

|  02152 

46 

15 
16 

338753 
344504 

959 

999897 
999894 

.0 
0.5 

OK 

333856 
344610 

959 

661144 
655390 

02181 

0-2211 

99976  45 
99976  44 

17 

350181 

QQA 

999891 

-O 

OK 

360289 

934 

649711 

02240 

99975 

43 

18 

355783 

Q99 

999888 

•  O 

OK 

355895 

922 

644105 

02269 

99974 

42 

19 
20 

361315 
366777 

910 

OQQ 

999885 
999882 

•O 

0.6 

Oc 

361430 
366895 

911 

OQQ 

638570 
633105 

02298 
02327 

99974 
99973 

41 
40 

21 

8.372171 

oyy 

OQQ 

9.999879 

.0 
OK 

8.372292 

oyy 

Quu 

11-627708 

02356 

9y972 

39 

22 

377499 

Ooo 

OryrTT 

999876 

•  O 

OK 

377622 

OOO 

0.70 

622378 

02385 

99972 

38 

23 

382762 

o77 

999873 

•  O 
OK 

382889 

o  /y 
867 

617111 

02414 

99971 

37 

24 

387962 

OKC 

999870 

.O 
OK 

388092 

857 

611908 

02443 

99970 

36 

25 

393101 

OOO 

Q4fi 

999867 

.O 
OK 

393234 

H17 

606766 

02472 

y9y69 

36 

26 

398179 

o4O 

999864 

•  O 
OC 

398315 

tj-±  t 

601685 

02501 

99969 

34 

27 
28 
29 
30 

403199 
408161 
413068 
417919 

827 
818 
809 

Him 

999861 
999858 
999854 
999851 

.O 

0.6 
0.5 

0.5 

Oil 

403338 
408304 
413213 
418068 

828 
818 
809 
800 

696662 
591696 
588787 
581932 

02530 
02560 
I  02589 
i  Oi-'blb 

99968!  33 
99967J  32 
99966  31 
SJ9966|  30 

31 

8.422717 

ouu 

nci  1 

9.999848 

•  O 
0£j 

8.422869 

7Q1 

11.677131  !  0264'1 

999G5!  29 

32 

427462 

/yi 

700 

999844 

•  O 
Oft 

427618 

/yjt 

783 

572382  ii  0261  (> 

99964 

!2b 

33 

432156 

iO  At 

774- 

999841 

.O 

Of] 

432315 

774 

667685 

02705 

999t>3 

27 

34 

436800 

i  /rt 

999838 

•  O 
0£» 

436962 

563038 

i  02V  34 

99963 

26 

35 
36 

441394 
445941 

758 

999834 
999831 

•  o 
0.6 

Oc 

441660 
446110 

758 

558440 
553890 

02/63 

!  02792 

99962J  25 
99961124 

37 

460440 

74.9 

999827 

.0 
Oc 

450613 

743 

549387 

02821 

99960 

23 

38 

454893 

1  %• 

999823 

.0 

Of? 

455070 

ryoK 

544930 

02850 

99959 

22 

39 

459301 

797 

999820 

•  O 
Oc 

469481 

/GO 

728 

640519 

02879 

99969 

21 

40 

463665 

i  A  I 

999816 

.  o 

Oc 

463849 

790 

536151 

0290b 

99968 

20 

41 

8.467985 

rj-t  o 

9.999812 

•  O 
Oc 

8.468172 

I  &\) 
71  3 

11.631828 

02938 

9995? 

19 

42 

472263 

/  L  w 
70fi 

999809 

•O 
O.c 

472454 

/  io 

707 

527546 

02967 

99956 

18 

43 

476498 

/UO 

/'(»<! 

999805 

O 
Oc 

476693 

/u/ 

523307 

1  0299u 

99955 

17 

44 

480693 

oyy 

999801 

.O 
Oc 

480892 

693 

619108 

03025 

99954  16 

45 

484848 

COC 

999797 

.O 

07 

485050 

686 

514950 

03054 

99953 

15 

46 

488963  I  ££ 

999793 

*  / 

On 

489170 

610830 

03080 

99952 

14 

47 

493040 

o/y 

999790 

•7 

Off 

493250 

on  A 

506750 

03112 

999621  13 

48 

497078 

673 

ccor 

999786 

.7 
07 

497293 

O/4 
fifiS 

602707 

03141 

999511  12 

49 

501080  i  SJ 

999782 

•  / 

Off 

501298 

OOO 
RG.1 

498702 

03170 

99950  11 

60 

606045  j  £ri 

999778 

.7 

e\  rt 

605267 

OO1 

494733 

03199 

99949 

10 

51 

8.608974  J"JJ 

9.99977f  f  X'X 

8.509200 

fi^O 

11.490800 

03228 

99948 

9 

52 

612867  XTJJ 

999769 

V  .  1 

613098 

VAA 

486902 

03257 

99947 

8 

53 

616726  1  £07 

999765 

0.7 

Oft 

616961 

544 

483039 

03286 

99946 

7 

54 

520551  -,' 

999761 

.7 
Off 

620790 

-oq 

479210 

03316 

9^946 

6 

55 

624343  J*" 

999767 

•  7 

624686 

«9^ 

476414 

03346 

99944 

6 

56 

628102  626 

999753 

0.7 

628349 

fi99 

471651 

03374 

99943 

4 

57 
58 

631828  ||f  J 
635523  °}? 

999748 
999744 

0.7 
0.7 

532080, 
535779 

616 

467920 
464221  1 

03403 
03432 

99942 
99941 

3 
2 

59 

539186  £n_ 

999740 

0.7 

539447 

611 

460553  |  034G1 

99940 

1 

60 

642819 

999735 

0.7 

643084 

606 

456916 

03490 

99939 

0 

Cosine. 

Sine. 

Cotanjc. 

Tanjr. 

N.  cos. 

N.sinc. 

' 

88  Degrees. 

TABLE  II.   Log.  Sines  and  Tangents.  (2°)  Natural  Sines.          23 

'  i   Sine. 

D.  10" 

Cosine. 

D.  10")  Tang. 

D.  10" 

Cotang.   |N.  sine.  IN.  eos 

018.642819 

fiflft 

9.999735 

0*7 

8.643084 

602 

11.456916  03490 

99939 

60 

1  |  546422 

ouu 

KQK 

999731 

.  / 

On 

646691 

RQ.'i 

453309!]  03519 

99938 

59 

2 

549995 

oyo 

999726 

•  t 
On 

560268 

oyo 

KQ1 

449732  |i03548 

99937 

58 

3 

653539 

OJl 

999722 

•  I 

553817 

oyi 

446183  I1  03577 

:)'):;:;<; 

57 

4 

657054 

586 

999717 

0-8 

657335 

587 

442664:!  03306 

93335 

56 

5 
6 

560540 
563999 

681 
576 

999713 
999708 

0-8 
0-8 

00 

580828 
684291 

582 
577 

Krt-Q 

439172  i!03635ly9934 
436709!  0366499933 

55 
54 

7 

667431 

Rfl7 

999704 

.O 

OQ 

667727 

O/o 

432273  (03693 

9993-2 

53 

8 
9 

570836 
574214 

563 

KKO 

999699 
999694 

-O 

0-8 

571137 
574520 

664 

428863!  03723 
425480  08752 

99931 
99930 

52 
51 

10 

577566 

ooy 

KK  A 

999689 

0-8 

00 

577877 

659 

422123JJ037S1 

99929 

50 

11 

8.580892 

OOrt 

KCA 

9.999685 

•  O 

8.581208 

655 

11.  418792;'  03810 

99927 

49 

12 
13 

684193 
687469 

uoU 
546 
642 

999680 
999675 

0-8 
0-8 

OQ 

584514 
687795 

551 
547 

4154S6  0383999926 
412205  |03868|99925 

48 
47 

14 

590721 

coo 

999670 

.O 

OQ 

691051 

Kon 

408949  03897  99924 

46 

15 

593948 

Ooo 

p.r)A 

999665 

.8 

0O 

694283 

boy 

KOK 

405717  !  03925 

99923 

45 

16 

697152 

00** 
p-QA 

999660 

.8 

597492 

OOO 

402508 

03955 

y9922 

44 

17 

600332 

OoU 
F>9fi 

999655 

0.8 
00 

600677 

631 

399323 

03984 

99921 

43 

18 

603489 

O-4O 

999650 

.0 

OQ 

603839 

KOQ 

396161 

04013 

99919 

42 

19 

606623 

OAZ 

999645 

.8 

606978 

O&o 

393022 

04042 

99918 

41 

20 

609734 

619 

999640 

0.8 

610094 

619 

389908 

04071 

99917 

40 

21 

8.612823 

515 

9.999635 

0.9 

8.613189 

616 

11.386811 

04100 

99916 

39 

22 

615891 

511 

CAQ 

999629 

0-9 
On 

616262 

612 

Kno 

383738 

03129 

99915 

38 

23 

618937 

OUO 

999324 

•  9 

Of-L 

619313 

OUo 

KAK 

380687 

04159 

99913 

37 

24 

621962 

CA1 

999619 

•9 
0Q 

622343 

OUo 

KA1 

377657 

04188 

99912 

36 

25 

624965 

OU1 

4Q7 

999614 

•9 

OQ 

626352 

OU1 

374648 

04217 

99911 

35 

26 

627948 

*iy  / 

999608 

•9 

628340 

498 

371660 

04246 

99910 

34 

27 

630911 

494 

999603 

0-9 

0*-v 

631308 

495 

Ac\*t 

368692 

04276 

99909 

33 

28 

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24         Log.  Sines  and  Tangents.  (3°;  Natural  Sines.   TABLE  II. 

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TABLE  II.   Log.  Sines  and  Tangents.  (4°)  Natural  Sines. 

25 

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26          Log.  Sines  and  Tangents.  (3°)  Natural  Sines.   TABLE  II. 

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TABLE  II.     Log.  Sines  and  Tangents.  (6C)  Natural  Sines.           27 

' 

Sine. 

D.  10'1  Cosine.  D.  10" 

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Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

N.sine. 

i 

83  Degrees. 

18 


Log.  Sines  and  Tangents.  (7°)  Natural  Sines.     TABLE  U. 

i 
0 

Sine. 

DHI7 
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Cosine. 

D.  10";   Tai  13. 

D.  iu 

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996219 

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877652 

13139 

99133 

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34 

119519 

169 

996202 

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128317 

101 
1  ci 

876683 

13168 

99129 

26 

35 

120469 

168 

996185 

2.8 

124284 

101 

1  fit 

875716 

13197 

99125 

25 

36 

121417 

158 

996168 

2.8 

125249 

101 

1  fif\ 

874751 

13226 

99122 

24 

37 

122362 

158 

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2.8 

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873789 

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23 

38 

123306 

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124248 

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99110 

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996100 

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870913 

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20 

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9.126125 

150 

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9.996083 

2.8 
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1  eq 

10.869959 

j  13370 

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19 

42 
43 

127060 
127993 

150 

156 

996066 
996049 

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2.9 

130994 
131944 

ioy 

158 

1  KO. 

869000 
868056 

i  13399 
13427 

99098 
99094 

18 
17 

44 

128925 

156 

996032 

2.9 

132893 

lOo 

1  F^Q 

867107 

13450 

99091 

16 

45 

129854 

155 

996016 

2.9 

133839 

lOo 

1  K7 

866161 

13485 

99087 

15 

46 

164 

995998 

2.9 
2f\ 

134784 

10  / 

805210 

13514 

99083 

14 

47 

131706 

154 

995980 

.9 

135726 

157 

804274 

13543 

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13 

48 
49 

132630 
133551 

163 

1  CO 

995963 
995946 

2.9 
2r\ 

136667 
137605 

156 

IfSK 

803383 
802395 

13672 
13hOO 

JS076 

19071 

12 
11 

60 

134470 

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995928 

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138642 

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99067 

10 

51 

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9.139476 

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10.  860624  i!  13658 

99063 

9 

62 

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995894 

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140409 

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859591  1  18681 

39059 

8 

53 

137216 

152 

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141340 

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858660  1371  G 

39055 

7 

54 

138128 

152 

995859 

2.9 

142269 

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867731  i  13744 

39051 

6 

65 

139037 

162 

995841 

2.9 

143196 

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5 

56 

139944 

151 

995823 

2.9 

144121 

1  £\A 

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67 

140850 

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145044 

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1  c«5 

864966  13831 

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3 

68 

141754 

161 

995788 

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145966 

loo 

1  rco 

854034  115860 

99035 

2 

59 

142655 

160 

995771 

2Q 

146885 

lOo 

1  KO 

853115 

13889 

99031 

1 

60 

143555 

150 

996753 

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147803 

1OO 

852197 

13917 

99027 

0 

Cosine. 

Sine. 

Cotang. 

Tang.   i  N.  cos. 

N.sine. 

' 

82  Degrees. 

TABLE  H.     Log.  Sines  and  Tangents.  (8°)  Natural  Sines. 

29 

' 

Sine. 

1).  10" 

Cosine. 

D.  10" 

Tang. 

JJ.  iu' 

Cotaug. 

A.  sine. 

.N.  cos. 

o 

9.143555 

9.995753 

3f\ 

9.147803 

10.852197 

13917 

99027 

60 

1 

144453 

150 

995735 

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3  A 

148718 

153 

851282 

13946 

99023 

59 

2 

145349 

149 

995717 

.0 

3  A 

149632 

152 

1  KO 

850368 

i  13975 

99019 

58 

3 

146243 

149 

995699 

.  U 

3  A. 

150544 

Io2 

849456 

'14004 

99016 

57 

4 

147136 

149 

995681 

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161454 

162 

848546 

14033 

99011 

56 

6 

148026 

148 

995664 

3.0 

3   A 

152363 

151 

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847637 

i  14061 

99006 

55 

6 

148915 

148 

995646 

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153269 

lol 

846731 

14090 

99002 

54 

7 

149802 

148 

995628 

3.0 

3  A 

164174 

151 

845826 

!  141  19  98998 

53 

8 

150686 

147 

995610 

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3  A 

156077 

150 

1  Kfl 

844923 

1  14148  98994 

52 

9 
10 
11 
12 

151569 
152451 
9.153330 
154208 

147 
147 
147 
146 

996591 
995573 
9.995555 
995537 

•  U 

3.0 
3.0 
3.0 

3  A. 

155978 
156877 
9.157776 
158671 

1DU 

150 
150 
149 

1  AQ 

844022 
843123 
10  842225 
841329 

•  14177  98990 
1420598986 

14263  98978 

51 

50 
49 

48 

13 

155083 

146 

995619 

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3f\ 

169565 

149 

840435 

!  14292 

98973 

47 

14 

155957 

146 

995601 

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160457 

149 

1  .(O 

839543 

i  14320 

98969 

46 

15 

156830 

146 

995482 

3.  1 
3-1 

161347 

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.  jQ 

838653 

i  14349 

98965 

45 

16 

157700 

145 

995464 

.  1 

34 

162236 

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1  AQ 

837764 

i  14378 

98961 

44 

17 

158569 

145 

995446 

.  1 
3t 

163123 

14o 

836877 

i  14407 

98957 

43 

18 

159435 

144 

995427 

.  1 
3-1 

164008 

1  /IT 

835992 

i  14436 

98953 

42 

19 
20 

160301 
161164 

144 
144 

995409 
995390 

.  1 

3.1 

31 

164892 
165774 

14/ 
147 

1  A'J 

835108 
834226 

!  14464 
i  14493 

98948 
98944 

41 
40 

21 

9.162025 

144 

9.995372 

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31 

9.166654 

14/ 

10.833346 

1  14522 

98940 

39 

22 

162885 

143 

995353 

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167532 

146 

832468 

14551 

98936 

38 

23 

163743 

143 

995334 

3.  1 
31- 

168409 

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14fi 

831591 

14580 

98931 

37 

24 

164600 

143 

995316 

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169284 

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830716 

!  14608 

98927 

36 

25 

165454 

142 

995297 

3.  1 

31 

170157 

145 

1  An 

829843 

14637 

98923 

35  ! 

26 

166307 

i  10 

995278 

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q  i 

171029 

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14ri 

828971 

14666 

98919 

34 

27 

167159 

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995260 

O.I 

31 

171899 

JL4O 

1  A  Pi 

828101 

14695 

98914 

33 

28 

168008 

142 

995241 

•  1 

39 

172767 

140 

1  A  A 

827233 

14723 

98910  32 

29 

168856 

141 

995222 

** 

30 

173634 

144 

1  4  A 

826366 

14752 

98906 

31 

30 

169702 

141 

995203 

•  He 

30 

174499 

144 

825501 

14781 

98902 

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31 
32 

9.170547 
171389 

141 
140 

1  /I  A 

9.995184 
995165 

•  <& 

3.2 

30 

9.175362 
176224 

144 

10.824638 
823776 

14810 
14838 

9b897 
98893 

29 

28 

33 

172230 

14U 

995146 

.  M 

o  o 

177084 

1  /1Q. 

822916 

14867 

98889 

27 

34 

173070 

140 

1  A  f\ 

9951271  I'* 

177942 

l^bo 

143 

822058 

14896 

J8884 

26 

35 

173908 

14U 

995108 

3.2 

178799 

142 

821201 

'•  14'o'26 

98S80 

25 

36 

174744 

139 

995089 

3'  9 

179655 

1  49 

820345 

'  14954 

J8876 

24 

37 

175578 

139 

i  on 

995070 

.  M 

39 

180503 

L*k/6 

81  9492  H14982 

J8871 

23 

38 

176411 

loy 

i  on 

995051 

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0  O 

181360 

142 

818640 

i  15011 

98867 

22 

39 

177242 

loy 

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995032!  *•„ 

182211 

141 

817789 

!  15040 

98863 

21 

40 

178072 

loo 

995013  5'o 

183059 

14:1 

816941 

I  15061! 

9h858 

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41 

9.178900 

138 

1  OQ 

9.994993  X*« 

9.183907 

141 

10-816093 

1509i 

;.>0fc64  i  1  •> 

4-J 

179726 

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994974  ;•; 

184752 

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815248 

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188120 

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811042 

1  15270 

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48 

184651 

136 

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3.3 

30 

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810-206 

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12 

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10-807706 

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134 
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137 

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136 

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0 

Cosine. 

Sine.  __  | 

Cotang. 

*  Tang. 

N.  cos. 

N.sine 

' 

81  Degrees. 

Log.  Sines  and  Tangents.  (9°)  Natural  Sines.     TABLE  II. 

' 

Sim*. 

D.  10 

Cosine. 

D.  lu 

Tung. 

IX  lu 

Cotang.  |,N.  sinc.j.S.  I'D,-'. 

0 

9.194332 

1  °^ 

9.994620 

3q 

9.199713 

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55 

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50 

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10.791881 

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789780 

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129 

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9.994191 

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1 

60 

239670 

119 

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3.7 

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123 

753681  ||  17365 

98481 

0 

Coaine. 

Sine. 

Cotang. 

Tang.   IJ  N.  cos. 

S'.sine. 

' 

80  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (10°)  Natural  Sines.          31 

, 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Co  tang. 

N.sine. 

N.  cos. 

0 

9.239670 

1  1Q 

9.993361 

37 

9.246319 

1  OQ 

10.753681 

17366 

98481 

60 

1 

240386 

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993329 

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247057 

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752943 

17393 

98476 

69 

2 

241101 

119 

993307 

3.7 

247794 

123 

752206 

17422 

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68 

3 

241814 

119 

993285 

8.7 

248630 

123 

761470 

17451 

98466 

67 

4 

242526 

119 

993262 

3.7 

249264 

122 

750736 

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98461 

56 

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243237 

118 

1  1  O 

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3.7 

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32          Log.  Sines  and  Tangents.  (11°)  Natural  Sines.     TABLE  IL 

0 

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TABLE  II.     Log.  Sines  and  Tangents.  (12°)  Natural  Sines. 

33 

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D.  10" 

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989413 

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47 

349329 

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24 

37 

339306 

94  0 

989384 

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47 

349922 

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650078 

21843  97585 

23 

38 
39 

339871 
340434 

93.9 

989356 
989328 

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4.7 

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360514 
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98.6 

AQ  K. 

649486 
648894 

21871  97579 
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21 

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340996 

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yo.o 

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20 

41 
42 

9.341658 
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9.989271 
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4.7 

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9.352287 
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10.647713  2195697560 
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18 

43 

342679 

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989214 

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353465 

yo  .  j 

QQ  n 

646535  i  22013  97547 

17 

44 

343239 

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989186 

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645947  122041 

97541 

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45 

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47 

354640 

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645360  2207097534 

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46 

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989128 

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40 

355227 

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47 

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Q7  fi 

644187  22126 

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07  *. 

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10.641861  '  22240  9  J49U  9 

62 

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Q(j_t 

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56 

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40 

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688947:  2i3S2!9,  463 

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57 

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69 

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91.4 

988782 
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1  ! 

60 

352088 

91  .3 

988724 

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363364 

yo  .  j 

686636  !|  22446 

97437 

0  :j 

Cosine. 

Sine. 

Cotang. 

Tang.  .   N.  cos. 

l\.3ine. 

' 

77  Degrees. 

34          Log.  Sines  and  Tangents.  (13°)  Natural  Sines.     TABLE  U. 

~0 

Sine. 

D.  10" 

Cosine. 
9.988724 

D.  10" 

Tang. 
9.363364 

D.  10"|  Cotang.  : 
^  JlO.  636636! 

N.&ine 

M.  cos. 

9.352088 

22495 

97437 

60 

1 

352635 

91.1 

988695 

4.9 

363940 

96.0 

636060  ;:  22523 

97430 

59 

2 

353181 

91.0 

988666 

4.9 
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364616 

95.9 

f\K  O 

635485  j!  22652 

97424 

58 

3 

353726 

90.9 

988636 

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4O, 

365090 

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634910  i  22580 

97417 

57 

4 

354271 

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988607 

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4  A 

365664 

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634336  '22608 

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56 

6 

354816 

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988578 

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366237 

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633763  122637 

97404 

65 

6 

355358 

90.5 

988548 

4.9 

366810 

95.4 

633190  '22665 

97398 

54 

7 

355901 

90.4 

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367382 

95.3 

632618  22693 

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53 

8 

356443 

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988489 

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367953 

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632047  |  22722 

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9 

356984 

90.2 

988460 

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3U8524 

95.1 

631  476  i  |22750 

97378 

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10 

357624 

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988430 

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369094 

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680906  '22778 

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11 

9.358064 

oy  .y 

9.988401 

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4Q 

9.369663 

94  .y 

10.630337  22807 

97365 

49 

12 
13 
14 
15 

358603 
359141 
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360215 

89.8 
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988371 
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4.9 
4.9 
6.0 

5  A 

370232 
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94.6 
94.6 
94.4 

629768  2283597368 
629201  I22863!97351 
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628067  1122920  973^8 

48 
47 
46 
45 

16 

360752 

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QA  O 

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372499 

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627601 

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97331 

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17 

361287 

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988223 

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5  A 

373064 

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18 

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6  A 

373629 

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626371 

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97318 

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362356 

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374193 

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625807 

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21 

9.363422 

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9.988103 

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9.375319 

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10.624681 

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22 

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375881 

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624119 

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23 

364485 

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988043 

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366016 

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377003 

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r\O  Q 

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25 

365546 

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987983 

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622437 

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26 

366076 

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OQ   1 

987953 

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378122 

93.2 

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27 

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987922 

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28 

367131 

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987892 

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5  A, 

379239 

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620761 

23288 

97261 

32 

29 

367659 

o7.y 

987862 

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5n 

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92.9 

620203 

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30 

368185 

87.7 

017  c 

987832 

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380354 

92.8 

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619646 

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9.368711 

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9.987801 

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9.380910 

y*i  / 

10.619090 

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Qrr  A 

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987710 

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617425 

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97210 

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37 

370808 
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87.1 
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987679 
987649 
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5.1 

6.1 
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383129 
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92.3 
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616871 
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97203 
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24 
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372373 

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609730 

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49 

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390815 

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609185 

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97106 

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50 

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987217 

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50 

391360 

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608640 

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10 

51 

9  379089 

85.4 

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9.987186 

iSi 

50 

9.391903 

90.6 
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10.608097 

23938 

97093 

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379601 

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607553 

23966 

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53 

380113 

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987124 

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60 

392989 

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607011 

23995 

97079 

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54 

380624 

85.1 

987092 

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393531 

yu.o 

606469 

24023 

97072 

6 

56 

381134 

86.0 

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5.2 

394073 

90.2 

605927 

24061 

97065 

6 

56 

381643 

84.9 

987030 

5.2 

394614 

90.1 

605386 

24079 

97058 

4 

57 

382152 

84.8 

986998 

5.2 

395154 

90.0 

604846 

24108 

J7051 

3 

68 

382661 

84.7 

986967 

6.2 

395694 

89.9 

604306 

24136 

97044 

2 

59 

383168 

84.6 

986936 

5.2 

396233 

89.8 

603767 

24164 

97037 

1 

60 

383675 

84.5 

986904 

5.2 

396771 

89.7 

603229 

2419i 

97030 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

N.sine. 

"  ' 

76  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (14°)  Natural  Sines 

35 

' 

Sine. 

D.  1U" 

Cosine. 

D.  i<y 

Tang. 

D.  10' 

Cotang. 

N.  sine 

IN.  COB.I 

0 

9.883675 

QA   A 

9.986904 

5f> 

9.396771 

on  c 

10.603229 

i  24192 

97030  60 

1 

384182 

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Q/i  O 

986873 

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397309 

oy.o 

602691 

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97023  59 

2 

384687 

o4.o 
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986841 

5.3 

60 

397846 

89.6 

on  c 

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j  24249 

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3 

385192 

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601617 

1  24277 

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5 

385697 
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81.0 

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986778 

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5.3 

50 

398919 
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55 

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386704 

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oo  o 

986714 

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50 

399990 

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985843 
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59 

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427547 

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25854 

96600 

1 

60 

412996 

lo  .O 

984944 

0. 

428052 

o4.o 

671948 

25882 

96593 

0 

Cosine. 

Sine. 

Co  tang. 

Tang. 

N.  cos. 

N.sint!. 

' 

76  Degrees. 

36          Log.  Sinea  and  Tangents.  (15°)  Natural  Sines. 

TABLE  II. 

' 

Sine. 

D.  10" 

Cosine. 

D.  10")   Tang. 

D.  10" 

CotHUg. 

N.  sim.v 

N.  co.- 

0 

9.412996 

70  c 

9.984944 

,  „  9.428052 

CM  0 

10.571948 

25882 

965(>3 

.., 

1 

413467 

/o  .0 

70  A 

984910 

D  •  I 

57 

428557 

OTt.  * 

571443 

259M 

-J658- 

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2 

413938 

to  .  4 

f70  o 

984876 

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5m 

429062 

O/i  A 

570938 

2593^ 

Jo57. 

5.-) 

3 

414408 

7o  .0 

984842 

,  7 
67 

429566 

O4.U 

S3  Q 

570434 

25  9ii- 

Jbi)  i  U 

•t~i 

4 

414878 

iO  .0 

984808 

.  / 

430070 

oo  .  i? 

569930 

2599  . 

9656"- 

5ti 

6 

415347 

78.2 

984774 

5.7 

430573 

83.8 

5694-27 

3602.I9655. 

o.- 

6 

415815 

78.1 

984740 

6.7 

431075 

83  .8 

on  >7 

568925 

26050  9!,»54< 

04  i 

7 

416283 

78.0 

1717  n 

984706 

5.7 

517 

431677 

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53 

8 

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77  P. 

984672 

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57 

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417217 

i  /  .0 

177  7 

984637 

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57 

432580 

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QO  A 

667420 

26135 

96524 

61 

10 
11 

417684 
9.418150 

1  1  .  I 

77.6 

77  K 

984603 
9.984569 

.  / 

5.7 

57 

433080 
9.433680 

Oo  .  *t 

83.3 

oo  O 

566920 
10.566420 

26163 
26191 

96517 
96509 

60 
49 

12 

418615 

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984536 

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5IM 

434080 

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co  O 

685920 

i  26219 

96602 

48 

13 

419079 

77.4 

984600 

.7 

434679 

bo.^ 

565421 

26247 

96494 

47 

14 

419544 

77.3 

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984466 

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435078 

83.1 

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564922 

26275 

96486 

46 

15 

420007 

77  .0 

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984432 

6.7 

436676 

OO  .  II 

564424  !  26308 

96479 

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16 

420470 

77  .* 

984397 

6.8 

50 

436073 

80  8 

663927 

26331 

96471 

44 

17 

420933 

77.  1 

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50 

436670 

oo  8 

653430 

j  26359 

96463 

43 

18 

421395 

77  .U 

984328 

.8 

50 

437067 

02.0 

80  o 

56-2933 

26387 

96456 

42 

19 

421857 

76  .9 

no.  Q 

984294 

.8 

50 

437663 

2  .  / 

562437 

26415 

96448 

41 

20 

422318 

76.O 

170  17 

984259 

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50 

438059 

89  'fi 

561941 

26443 

96440 

40 

21 

9.422778 

76.  / 

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9.984224 

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9.438654 

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1  0.  56  1  416  |!  26471 

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22 

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984190 

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Bo 

439048 

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26500 

96425 

98 

23 

43365*7 

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439543 

82.0 

560457 

26628 

96417 

37 

24 

424156 

76.6 

984120 

6.8 

5Q 

440036 

82.0 
H2  2 

659964 

1  26556 

96410 

36  1 

25 
26 

27 
28 

424615 
425073 
426530 
425987 

76^3 
76.2 
76.1 

984086 
984050 
984016 
983981 

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6.8 
6.8 
6.8 

50 

440629 
441022 
441614 
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82  .'l 
82.0 
81.9 

HI  9 

559471  !  26584 
658978  126612 
558486  26640, 
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96402 
96394 
96386 
96379 

35 
34 
33 
32 

29 
30 

426443 
426899 

7e!o 

983946 
983911 

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5.8 

442497 
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ol  .  & 

81.8 

657503 

557012 

26696 
1  26724 

96371 
96363 

31 
30 

31 
32 
33 

9.427354 
427809 
428263 

75.9 

75.8 
75.7 

9.983875 
983840 
983805 

6.8 
6.8 
6.9 

9.443479 

443968 
444458 

81  .7 
81.6 
81.6 

wi  f* 

10.556521 
556032 
555542 

j  26752 
!  26780 
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96365 
96347 
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29 

28 
27 

34 

428717 

75  .6 

983770 

5.9 

444947 

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555053  2t>83t> 

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26 

35 

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24 

37 

38 

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983664 
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446411 

446898 

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26948 

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430978 

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431429 

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76.0 

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5.9 

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18 

43 
44 

432778 
433226 

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983452 
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550u74 
550190 

270bb 
27  lib 

Jb2Gl 

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16 

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433675 

74.7 

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5.9 
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450294 

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27144 

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15 

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434122 

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983345 

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549223 

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434569 

74.5 

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548  .'40 

27200 

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74.4 

983273 

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983094 

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454148 

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545852 

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7  j, 

54 

437686 

74.0 

983058 

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79.9 
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56 

438572 

73.8 

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70  7 

544414 

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4 

57 

439014 

73  .7 

982960 

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t-Q  f^ 

643936 

27480 

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58 

439456 

73.6 

982914 

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456642 

t  y  .  o 

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2760b 

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2 

59 

439897 

73  .6 

982878 

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642981 

27636 

^b!34 

1 

60 

440338 

73.6 

982842 

6.0 

457496 

i9.o 

642504 

27564 

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Sine. 

Cotaug. 

Tang. 

>;.  cos.)  \.SIKO. 

~~r 

74  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (16°)  Natural  Sines.         37 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang. 

N.  sine. 

N.  cos. 

o 

9.410338 

9.982842 

9.467496 

rtf\  A 

10.642504 

27664 

96126 

60 

1 

440778 

73.4 

7Q  Q 

982805 

6.0 
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457973 

79.4 

rjr\  o 

542027 

27592 

96118 

59 

2 

441218 

lOtO 

170  o 

982769 

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458449 

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7Q  ^ 

641551 

27620 

96110 

68 

3 

441658 

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982733 

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458925 

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27648 

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982696 

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459400 

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640600 

27676 

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6 

442635 

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982660 

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469875 

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rt(\  f\ 

640125 

27704 

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442973 

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982624 

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460349 

79  .U 

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639651 

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54 

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443410 

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460823 

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463186 

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446459 

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464129 

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27955 

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464699 

78.4 

635401 

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447326 

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465069 

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95997 

44 

17 

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982220 

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465539 

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28039 

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627468  1 

28467 

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33 
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454619 
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70.9 
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6.3 
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472996 
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77.1 
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527005 
626543 

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95857 
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27 
26 

35 

455469 

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6.3 

60 

473919 

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981512 

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525619 

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95832 

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37 

456316 

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Sine.  | 

Cotang. 

Tiin^r.   ;A.C-O?. 

N'.fiine. 

/ 

73  Degrees. 

Log.  Sines  and  Tangents.    (17°)    Natural  Sines.           TABLE  II. 

Sine. 

D.  10 

"     Cosine. 

D.  1 

Tang. 

D.  10 

"     Cotang.      N.  sine 

.N.  co 

9.46593 

68  f 

9.980596 

6t 

9.48533 

10,514661 

1  2923 

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2 

59 

633704 

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3979 

1 

60 

634052 

57.9 

972986 

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561066 

5.6 

438934  1|  34202 

3909 

0 

Cosine. 

Sine. 

Cotang. 

Tang.   UN.  cos. 

.sine. 

~7 

70  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (20°)  Natural  Sines.          41 

' 

Sine. 

P.  10" 

Cosine. 

D.  10' 

Tasg. 

D.  10" 

CuLaug.   N.  sine. 

N.  cos. 

o 

9.534052 

rrt  o 

9.972986 

9.5(>106(> 

0~   K 

10.438934  3420-2 

93969 

60 

1 

634399 

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bo  .5 

438511  '  34229 

93959 

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2 

634745 

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972894 

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7nr 

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438  149;  '3425  7 

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3 

535092 

67.7 

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562244 

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93939 

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4 

635438 

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437364;  34311 

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5 

535783 

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6 

636129 

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12 

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640249 

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37 

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642293 

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643987 

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10.426877; 

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970827 

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2 

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583800 

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35810 

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584177 

415823 

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93368 

0 

Cosine. 

Sine. 

Cotang. 

Tang,   I  N.  cos;. 

N.siuo. 

~T~ 

C9  Degrees. 

Log.  Sines  and  Tangents.  (21°)  Natural  Sines. 

TABLE  H. 

Sine. 

D.  10 

Cosine. 

D.  10 

Tang. 

D.  10"i  Cotang. 

;N  .sine 

N.  cos.  | 

0 

9.654329 

KA   Q 

9.970152 

81 

9.584177 

6o  9  10.415823 

!  35837 

93358  00 

3 

654658 
664987 
555315 

O4.  0 

54.8 
54.7 
54  7 

970103 
970055 
970006 

.  J 
8.1 
8.1 
8  1 

584555 
684932 
585309 

62^9 
62.8 
62  8 

415445  13586^ 
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414091  |-3591fc 

93348 
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58 
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413938  1  3597L 

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969860 

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81 

586439 

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68  Degrees.                          II 

TABLE  IT.     Log.  Sines  and  Tangents.  (22°)  Natural  Sines.          43 

~~r~ 
0 

Sine. 
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Tang. 

X.  cos.JN.sine. 

' 

67  Degrees. 

19 


Log.  Sines  and  Tangents.  (23°)  Natural  Sines.     TABLE  II. 

' 

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D  10' 

Cosine. 

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47 

605606 

4/  -8 

j  rr  O 

961458 

.  o 

90 

644148 

j'  .  1 

^T  A 

355862 

40328 

91508 

13 

48 

605892 

47.0 
47  7 

961402 

BO 

9  3 

644490 

oi  .0 
S7  rt 

355510 

40355 

91496 

12 

49 

606179 

4/  «  / 

47  7 

961346 

93 

644832 

j  t  ,  \j 

S7  O 

356168 

40381 

91484 

11 

50 

606465 

4*  •  / 
47  fi 

961290 

9*3 

645174 

5  *  .  U 
66  Q 

354826 

40408 

1472 

10 

51 

9.606751 

4/  •  O 
47  R 

3.961235 

Q*3 

9.645516 

>u  .  y 
56  9 

10.354484 

40434 

91461 

9 

52 

607036 

4/  »D 

47  ft 

961179 

y  .  o 

9  3 

645857 

S6  Q 

354143 

40461 

J1449 

8 

53 

607322 

4  1  «O 

961123 

9*  q 

646199 

jo  .y 
FiK  Q 

353801 

40488 

1437 

7 

54 

607607 

47-5 

47  K 

961087 

.  O 

9  3 

646540 

oo  .y 
56  8 

363460 

40614 

1426 

6 

55 

607892 

4i  •  O 

47  4 

961011 

93 

646881 

56  8 

353119 

40541 

1414 

6 

56 

608177 

4/  •  4 

/iO'  A 

960955 

9"o 

647222 

-C  j3 

352778 

40667 

1402 

4 

57 

608461 

47  .4 

Xnr  >< 

960899 

.  o 
9q 

647562 

3O  .  o 

•C  ft 

352438 

40594 

1390 

3 

58 

608745 

47  .4 

/iT  Q 

960843 

,  o 

9  A 

647903 

DO.  / 

'£•  7 

352097 

40621 

1378 

2 

59 

609029 

47  .0 

960786 

.4 

648243 

DO.  / 

K£»  ry 

351767 

40647 

1366 

1 

60 

609313 

47.3 

960730 

9.4 

648583 

OO.7 

351417 

40674 

1355 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

\.sine. 

' 

66  Degrees. 

TABLE  IT.     Log.  Sines  and  Tangents.  (24°)  Natural  Sines.          45 

' 

Sine. 

D.  10 

Cosine. 

D.  1U 

Tang. 

D.  10 

Uotang. 

N.  sine 

N.  cos 

0 

9.609313 

\fi  O 

9.960730 

9  A 

9.648583 

~C  f! 

10.351417 

40674 

91355 

60 

1 

609597 

4  I  .  O 

47  2 

960674 

.  4t 

94 

648923 

oo  .  o 

fifi  fi 

351077 

40700 

91343 

69 

2 

609880 

4  /  .  *> 

960618 

.  ft 

649263 

5D  .  C 

350737 

40727 

91331 

58 

3 

610164 

47.2 

960561 

9.4 

649602 

56.6 

350398 

40753 

91319 

57 

4 

610447 

47.2 

960505 

9.4 

649942 

56  6 

350058 

40780 

91307 

56 

5 

610729 

47.1 

960448 

9.4 

650281 

56.5 

349719 

40806 

91295 

55 

6 

611012 

47.1 

4^7  A 

960392 

9.4 

650620 

56.6 

349380 

40833 

91283 

54 

7 

611294 

47.  U 

trj  n 

960335 

9.4 

9    A 

650959 

59.6 

349041 

40860 

91272 

53 

8 

611576 

47.  U 

Art  A 

960279 

.4 

9    A 

661297 

66.4 

e£l  A 

348703 

40886 

91260 

52 

9 

611858 

47  .  U 

AC  A 

960222 

.4 

94. 

651636 

6D.4 
Kfi  A 

348364 

40913 

91248 

51 

10 

612140 

40  .  y 

AC*  O 

960165 

.  *t 

651974 

50.  4 

K£>   O 

348026 

40939 

91236 

50 

11 

9.612421 

4o.y 

9.960109 

9.4 

9.652312 

OO.O 

10.347688 

40966 

91224 

49 

12 

612702 

46.9 

AC  Q 

960052 

9.6 

652650 

66.3 

347350 

40992 

91212 

48 

13 

612983 

4o.o 
Afi  8 

959995 

9.5 
9K 

652988 

56.3 

C£!  o 

347012 

41019 

91200 

47 

14 

613264 

4O.O 

959938 

.0 

653326 

5b.o 

346674 

41045 

91188 

46 

15 

613545 

46.  7 

959882 

9.5 

653663 

56.2 

346337 

41072 

91176 

46 

16 

613825 

46.  7 

959825 

9.5 

654000 

56.2 

346000 

41098 

91164 

44 

17 

614105 

46.  7 

959768 

9.5 

654337 

56.2 

345663 

41126 

91152 

43 

18 

614385 

46.6 

.4r>  r* 

959711 

9.5 

654174 

56.1 

345326 

41161 

91140 

42 

19 

614665 

4o.o 

959654 

9,5 

655011 

66.1 

344989 

41178 

91128 

41 

20 

614944 

46.6 

AC1  K 

959596 

9,5 

655348 

66.1 

344652 

41204 

91116 

40 

21 

9.615223 

4o.  O 

9  959539 

9,6 

9.655684 

56.1 

10.344316 

41231 

91104 

39 

22 

615502 

46.6 

Ar*  e- 

959482 

9.6 

656020 

66.0 

343980 

41257 

91092 

38 

23 

615781 

4o.O 

959425 

9.5 

656356 

56.0 

343644 

41284 

91080 

37 

24 

616060 

46.4 

959368 

9.6 

656692 

56.0 

343308 

41310 

91068 

36 

25 

616338 

46.4 

Ad  A 

959310 

9.6 
9f» 

657028 

56.9 

K"  O 

342972 

41337 

91056 

36 

26 

616616 

4O.4 

959253 

.0 

657364 

60.  y 

342636 

41363 

91044 

34 

27 

616894 

46,3 

A&  O 

959195 

9,6 

657699 

55.9 

342301 

41390191032 

33 

28 

617172 

4o.d 

AG  O 

959138 

9.6 

658034 

55.9 

341966 

41416 

91020 

32 

29 

617450 

4t>.!s 

Art  r) 

959081 

9.6 

668369 

55.8 

341631 

41443 

91008 

31 

30 

617727 

4O..2 

959023 

9.6 

668704 

55.8 

ee  Q 

341296 

41469 

90996 

30 

31 

9.618004 

46.2 

9.958965 

9,6 

9,659039 

5o.o 

10  340961 

41496 

90984 

29 

32 

618281 

46.  1 

Aa  i 

958908 

9.6 

659373 

5o.o 

340627 

41522 

90972 

28 

33 

618568 

4o.  1 

968850 

9.6 

669708 

55,7 

340292 

41549 

90960 

27 

34 

618834 

46.1 

Ad  A 

958792 

9.6 

660042 

55.7 

339958 

41576 

90948 

26 

35 

619110 

4o,U 

Af>   /V 

958734 

9,6 

660376 

55.7 

339624 

41602 

90936 

26 

36 

619386 

4O.U 

Af?  A 

958377 

9.6 

660710 

55.7 

339290 

41628 

90924 

24 

37 

619662 

4b.  U 
4~  o 

958619 

9.6 

661043 

56.6 

338957 

41655 

90911 

23 

38 

619938 

4o.  y 

958561 

9,6 

661377 

55.6 

338623 

41681 

90899 

22 

39 
40 

620213 

620488 

45.9 
45.9 

A  K  O 

958503 
958445 

9.6 
9.7 

661710 
662043 

65.6 
65,6 

338290 
337957 

41707 
41734 

90887 
90876 

21 
20 

41 

9.620763 

45.  o 

AK.  O 

9.958387 

9,7 

9.662376 

55.5 

10.337624 

41760 

90863 

19 

42 
43 
44 

621038 
621313 
621587 

4o.o 

45.7 
45.7 

AS-   ~ 

958329 
958271 
958213 

9,7 
9.7 
9.7 

662709 
663042 
663375 

55,6 
55,4 
65,4 

337291 
336958 
336626 

41787  90851 
41813!90839 
41840190826 

18 
17 
16 

45 
46 
47 

621861 
622135 
622409 

45.7 
45.6 
45.6 

A&   £» 

958154 
958096 
958038 

9.7 
9.7 
9.7 

663707 
664039 
664371 

55.4 
56.4 
55.3 

336293 
335961 
335629 

41866190814 
4189290802 
41919)90790 

15 
14 
13 

48 

622682 

^10.  D 

Afi  e 

957979 

9.7 

664703 

65.3 

335297 

41945 

90778 

12 

49 

622956 

^5  .  o 
4f>  f^ 

957921 

9.7 

665035 

55.3 

K"  O 

334965 

41972 

90766 

11 

50 

623229 

^O  .  o 

957863 

9.7 

665366 

OO,O 

334634 

41998 

90753 

10 

51 

9.623512 

45.5 

9.957804 

9.7 

9.665697 

•55,2 

0.334303 

42024 

30741 

9 

52 

623774 

45.4 

957746 

9.7 

666029 

55.2 

333971 

42051 

J0729 

8 

53 

624047 

45.4 

Af   A 

967687 

9.8 

666360 

55.2 

333620 

42077 

20717 

7 

54 

624319 

45.4 

A  -  O 

957628 

9.8 

666691 

65.1 

333309 

42104 

)0704 

6 

55 

624591 

4o.O 

AK.  O 

957570 

9.8 

667021 

55.1 

332979 

42130 

90692 

5 

56 

624863 

4O.O 

957511 

9.8 

667352 

65.  1 

332648 

42156 

^0680 

4 

57 

625135 

46.3 

957452 

9.8 

667682 

55.1 

332318 

42183 

J0668 

3 

58 

625406 

45.2 

957393 

9.8 

668013 

55.0 

331987 

42209 

W655 

2 

59 

625677 

45.2 

957336 

•9.8 

668343 

65.0 

331657 

42235 

)0643 

1 

60 

625948 

45.2 

957276 

9,8 

668672 

56.0 

331328 

42262 

)0631 

0 

Cosine. 

Sine. 

Cotang.  1 

Tang. 

N.  cos.  |NT.  sine. 

' 

65  Degrees. 

46          Log.  Sines  and  Tangents.  (25°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

1).  10" 

Cosine.  |D.  10" 

Tang. 

D.  10' 

Cotang.  [(N.sine 

N.  cos. 

0 

9.625948 

45  1 

9.957276 

Q  ft 

9.668673 

KK  0 

10.331327 

42262 

90631 

60 

1 

626219 

957217 

y  .  o 

90 

669002 

oo.  u 

330998 

42288 

90613 

59 

2 

626490 

45.  1 

AK.   1 

957158 

.0 
90 

669332 

54.9 

f.A   Q 

330668  42315 

90606 

68 

3 

626700 

4O  .  1 

AK  A 

957099 

,  o 
9Q 

669661 

04.  y 

K.A  Q 

330339 

42341 

90594 

67 

4 

627030 

4O  .  \s 

AK.  0 

957040 

.  O 
9Q 

669991 

04.  y 
54  8 

330009 

42367 

90582 

66 

5 

627300 

4O  .  U 

A  K  A 

956981 

.  O 
90 

670320 

K4*R 

329680 

42394 

90569 

55 

6 

627570 

40  .  U 

A  A  Q 

956921 

.  o 
9q 

670649 

04,  o 

K.A   O 

329351  |i42420 

90557 

54 

7 

627840 

44,  y 

AA  Q 

956862 

.  y 

9Q 

670977 

O4.  o 

K.A  Q 

329023 

4244b 

90545 

63 

8 

628109 

44.  y 

AA  Q 

956803 

.  y 
9q 

671306 

04,  0 

f>4  7 

328694 

42473 

90532 

52 

9 

628378 

44.  y 
44  Pi 

956744 

.  y 
90 

671634 

04  .  / 

fi4  7 

328366 

42499 

90520  51 

10 

628647 

44.  o 

AA  Q 

958684 

.  y 
9q 

671963 

O4,  i 
fi4  7 

328037 

42525 

90507 

50 

11 

9.628916 

44.  O 
A  A  7 

9.956625 

,  y 
9  9 

9.672291 

O4,  / 

n4  7 

10.327709 

42552 

90495 

49 

12 

629185 

44.  * 

44  7 

956566 

9*9 

672619 

04.  i 
54  fi 

327381 

42578 

90483 

48 

13 

629453 

44,  / 

44  7 

956506 

99 

672947 

O4.  D 

54  6 

327053 

42604 

90470 

47 

14 

629721 

44  ,  / 

44  R 

956447 

99 

673274 

54  6 

326726 

42631 

90458 

46 

15 

629989 

44.  O 

44  6 

956387 

99 

673602 

54  6 

326398 

42657 

90446 

45 

16 

630257 

AA.  fi 

956327 

q  Q 

673929 

K.A   C 

326071 

42683 

90433 

44 

17 

630524 

44.  O 

44  6 

956268 

y  & 
9  9 

674257 

O4,  O 

54  5 

325743 

42709 

90421 

43 

18 

630792 

44  ft 

956208 

1  0°  0 

674584 

K4*f> 

325416 

42736 

90408 

42 

19 

631059 

44.  O 

44  Ft 

956148 

J.U  .  v 

100 

674910 

O4.  O 

P.A  A 

325090 

42762 

90396 

41 

20 

631326 

44.  0 

44  f» 

956089 

1U,  v 
TOO 

675237 

O4.  4 
K.A  A 

324763 

42788 

90383 

40 

21 

9.631593 

44  .  0 

44  4 

9.956029 

L\J  ,  v 

10  0 

9.675564 

O4.  4 

54  4 

10.324436 

42815 

90371 

39 

22 

631859 

44.4 

44  4 

955969 

10  0 

675890 

54  4 

324110 

42841 

90358 

38 

23 

632125 

44.  ft 

44  4 

955909 

10  0 

676216 

54*3 

323784 

42867 

90346 

37 

24 

632392 

44  3 

955849 

10*0 

676543 

54  3 

323457 

42894 

90334 

36 

25 

632658 

AA  Q 

955789 

in'  () 

676869 

K/1   O 

323131 

i  42920 

90321 

35 

26 

632923 

44.  o 

44  3 

955729 

J.U.  U 

10  0 

677194 

O4.  o 

54  3 

322806 

i  42946 

90309 

34 

27 

633189 

44  2 

955669 

10  0 

677520 

54*2 

322480 

42972 

90296 

33 

28 

633454 

44*2 

955609 

lo'o 

677846 

54*2 

322154 

42999 

90284 

32 

29 

633719 

44^2 

955548 

10  0 

678171 

64*2 

321829  43025 

90271 

31 

30 

633984 

44  1 

955488 

10  0 

678496 

KA  o 

321504 

43051 

90259 

30 

31 

9.634249 

44.  1 

44  1 

9.955428 

101 

9.678821 

04,  ^ 
F>4  1 

10.321179 

43077 

90246 

29 

32 

634514 

44.  i 

44  ft 

955368 

1  v,  A 

10  1 

679146 

04.  i 
54  1 

320854 

43104 

90233 

28 

33 

634778 

44,  u 

44  0 

955307 

lv.  J- 

10  1 

679471 

O4,  JL 
54  1 

320529 

43130 

90221 

27 

34 
35 

635042 
635306 

44,  U 

44.0 
43  Q 

955247 
955186 

1U,  A 

10.1 

10  1 

679795 
680120 

04,  i 
54.1 
54  0 

320205 
319880 

43156 
1  43182 

90208 
90196 

26 

25 

36 

635570 

40  ,  y 

4Q  Q 

955126 

10  1 

680444 

O4,  U 

54  0 

319556  |43209 

90183 

24 

37 

635834 

40  .  y 
43  Q 

955065 

10  1 

680768 

O4,  U 

54  0 

319232 

43235 

90171 

23 

38 

636097 

40  .  y 

43  8 

955005 

10*1 

681092 

O4,  U 

54  0 

318908 

43261 

90158 

22 

39 

636360 

43*8 

954944 

10  1 

681416 

O4.  U 

53  9 

318584 

1  43287 

90146 

21 

40 

636623 

43  'y 

954883 

io!i 

681740 

53  9 

318260 

43313 

90133 

20 

41 

9.636886 

AQ  n 

9.954823 

10  1 

9.682063 

=  O   Q 

10.317937 

43340 

90120 

19 

42 

637148 

4o  .  < 

43  7 

954762 

io!i 

682387 

oo  .  y 
53.9 

317613 

43366 

90108 

18 

43 

637411 

43  7 

954701 

101 

682710 

xq  u 

317290 

43392 

90095 

17 

44 

637673 

4o  .  / 

43,7 

954640 

L\J  .  *• 

10.1 

683033 

OO  ,  O 

53.8 

316967 

43418 

J0082 

16 

45 

637935 

40  c 

954579 

lo'i 

68335G 

Xq  Q 

316644 

43445 

90070 

15 

46 

638197 

4O  ,  O 

4*1  fi 

954518 

10*2 

683679 

Oo  .  O 
Kq  o 

316321 

!  43471 

90057 

14 

47 
48 

638458 
638720 

4o  .O 

43.6 

An  K 

954457 
954396 

10'.2 
102 

684001 
684324 

Oo  .  o 

53.7 

RO  •? 

315999 
315676 

43497 
i  43523 

90045 
90032 

13 
12 

49 

638981 

4O.  O 
40  c 

954335 

1U.  ^ 

10  2 

684646 

Oo.  / 

Kq  7 

315354 

i  43549 

90019 

11 

50 

639242 

4O  .  O 
4Q  K 

954274 

10  2 

684968 

Oo  .  / 

Kq  7 

315032 

43575 

90007 

10 

51 

9.639503 

4O  .  O 

4Q  A 

9.954213 

10*2 

9.685290 

Oo  .  / 

53  6 

10.314710 

43602 

89994 

9 

52 

639764 

4o  .  4 

40  4 

954152 

10*2 

685612 

53  6 

314388 

43628 

89981 

8 

53 

640024 

4o  .  4 

43  4 

954090 

10*2 

685934 

53  6 

314066 

43654 

89968 

7 

54 

640284 

40  .  *± 
43  3 

954029 

10  2 

686255 

53  6 

313745 

43680 

89956 

6 

55 

640544 

40  q 

953968 

10  2 

686577 

eq  e 

313423 

43706 

89943 

5 

56 

640804 

4o  .  o 
40  q 

953906 

10  2 

686898 

Oo  .  O 

rq  C 

313102 

43733 

89930 

4 

57 

641064 

4o  .  O 
40  O 

953845 

10  2 

687219 

Oo  ,  O 
eq  K 

312781 

43759 

89918 

3 

58 

641324 

4o  .  *f 
4q  o 

953783 

10  2 

687540 

Oo  ,  O 
cq  ft 

312460 

43785 

89905 

2 

59 

641584 

4o  .^ 
40  o 

953722 

10  3 

687861 

Oo  ,  O 
ccq  4 

312139 

43811 

89892 

1 

60 

641842 

4o  .  ^ 

953660 

688182 

Oo  .  4 

311818 

43837 

89879 

0 

Cosine.  I 

Sine. 

Co  tang. 

Tang.   ||  N.  co«. 

X.mno. 

~~f~ 

64  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (26°)  Natural  Sines.          47 

|   Sine.  |D.  10' 

Cosine. 

D.  10" 

Tang. 

D.  10' 

Cotang.   N.  sine 

N.  cos 

0 

9.641842 

4q  i 

9.953660 

103 

9.688182 

cq  4 

10.3118181  43837 

89879 

60 

1 

642101 

4o  .  i 

953599 

lu  .  o 

688502 

Oo.  4 

31  1498  i!  43863 

89867 

59 

2 

642360 

43.1 

,tn   1 

953537 

10.3 

688823 

53.4 

311177  ji  43889 

89854 

58 

3 

642618 

4o.  1 

43  0 

953475 

10.  3 
i  n  Q 

689143 

53  .4 

CO   O 

310857  43916 

89841 

57 

4 

642877 

4o  .  U 
43  0 

953413 

lU.o 

i  A  q 

689463 

Do  .  o 

CO  O 

310537  i  43942 

89828 

66 

6 

643135 

4o  .  U 
43  0 

953352 

IU.  o 
i  A  q 

689783 

Oo  .  o 

310217 

43968 

89816 

55 

6 

643393 

4o  ,  u 
43  0 

953290 

lu,  o 
1  0  3 

690103 

eq  q 

309897 

43994 

89803 

54 

r? 

643650 

4o  .  U 

42  9 

953228 

lu.  o 
103 

690-423 

Oo  ,  o 
cq  q 

309577  jj  44020 

89790 

53 

8 

643908 

953166 

IU.  o 
103 

690742 

Oo  .  o 

cq  9 

309258 

44046 

89777 

52 

9 

644165 

42  9 

953104 

lu.  o 
103 

691062 

OO.  6 

53  2 

308938 

44072 

89764 

51 

10 

644423 

49  8 

953042 

lu,  o 
in  Q 

691381 

308619 

44098 

89752 

50 

11 

9.644680 

4^  .  o 

9.952980 

lU.o 

9.691700 

Od.2 

10.308300 

44124 

89739 

49 

12 

644936 

42  8 

952918 

10.4 
1  O  4 

692019 

53.  1 

CO   1 

307981 

44151 

89726 

48 

13 

645193 

42  7 

952855 

lu.4 

692338 

DO  .  1 
cq  i 

307662 

44177 

89713 

47 

14 

645450 

952793 

104 

692656 

Oo  .  1 
cq  i 

307344 

44203 

89700 

46 

15 

645706 

40  7 

952731 

lu.  ^ 
1  A  4 

692975 

Oo  .  1 
cq  i 

307025 

44229 

89687 

45 

16 
17 

645962 
646218 

'it,  I 

42.6 

952669 
952606 

lu.  ^ 
10.4 

693293 
693612 

Do  .  1 

53.0 

•306707 
306388 

44255 
44281 

89674 
89662 

44 
43 

18 

646474 

42.6 

952544 

10.4 

1  A  A 

693930 

53.  0 

306070 

44307 

89649 

42 

19 

646729 

49  Fi 

952481 

10.4 
104 

694248 

53  .  0 

cq  A 

305752 

44333 

89636 

41 

20 

646984 

4,6.  O 

952419 

IU,  4 

694566 

Oo  .  U 

305434 

44359 

89623 

40 

21 

9.647240 

49  P> 

9.952356 

10.4 
104 

9.694883 

52.  9 
52  9 

10-305117 

44385 

89610 

39 

22 

647494 

4^,  O 

952294 

lu.  ^t 

695201 

304799 

44411 

89697 

38 

23 

647749 

42.4 

952231 

10.4 

695518 

62.9 

304482 

44437 

89584 

37 

24 

648004 

42.4 

952168 

10.4 

696836 

62.9 

304164 

44464 

89571 

36 

25 

648258 

42.4 

952106 

10.5 

696153 

52.9 
62  8 

303847 

44490 

89558 

35 

26 

648512 

40  Q 

952043 

10.  o 

698470 

303580 

44616 

89545 

34 

27 

648766 

49  3 

951980 

10.5 

1  AC 

696787 

52.  8 

52  8 

303213 

44542 

89532 

33 

28 

649020 

^±^.  o 

951917 

IU.  O 

697103 

302897 

44568 

89519  32 

29 
30 

649274 

649527 

42.3 

42.2 

951854 
951791 

10.6 
10.5 

697420, 
697736 

52.8 
52.7 

CO  rt 

302580 
302264 

44594 
44620 

89506  31 
89493  30 

31 

9.649781 

42.  2 

AO  o 

9.951728 

10.5 

1  O  fi 

9.698053 

0^.7 
52  7 

10-301947 

44646 

89480 

29 

32 
33 

650034  ^'" 
660287  ;**  7 

951665 
951602 

lu.  D 

10.6 
1  0  Pi 

698369 
698685 

52!? 

301631 
301315 

44672 
44698 

89467 
89454 

28 

27 

34 

650539 

951539 

IU.  O 

699001 

D-&.U 

300999 

44724 

89441 

26 

36 

650792 

42.1 

951476 

10.6 

699316 

52.6 

300684 

4475089428 

25 

36 

661044 

42.1 

951412 

10.5 

699632 

o~.6 

300368 

44776 

89415 

24 

37 

651297 

42.  0 

951349 

10.5 

699947 

o2.6 

"O  l? 

300058 

44802 

89402 

23 

38 

651649 

42.  0 

951286 

10.6 

700263 

3^.6 

299737 

44828 

89389  1  22 

39 

651800 

42  .  0 

951222 

10.6 

700678 

oA5 

299422 

44854 

89376  21 

40 

652052 

41.9 

951169 

10.6 

700893 

52.6 

299107 

44880 

89363 

20 

41 

).  652304 

41.9 

9  951096 

10.6 

9.701208 

52.6 

10-298792 

44906 

89360 

19 

42 

652555 

41.9 

At    Q 

951032 

10.6 

701523 

52.4 

298477 

44932 

89337 

18 

43 

652806 

41.8 

950968 

10.6 

701837 

KO   A 

298163 

44958 

89324 

17 

44 

653057 

41.8 

A-t    U 

950905 

10.6 

702152 

3^.4 

297848 

44984 

89311 

16 

45 

653308 

41.o 

950841 

10.6 

702466 

52.4 

297534 

45010 

89298 

15 

46 

653658 

41.8 

950778 

10.6 

702780 

52.4 

297220 

45036 

89285 

14 

47 

653808 

41.7 

950714 

10.6 

703095 

52.3 

296905 

45062 

89272 

13 

48 

654059 

41.7 

i  1  i 

950650 

10.6 

703409 

52.3 

296691 

46088 

89269 

12 

49 

654309 

41.  / 

950586 

10.6 

i  A  f: 

703723 

-n 

296277 

45114 

89245 

11 

50 

654558 

41.6 

950522 

10.  D 

704036 

5.2  .  3 

295964 

45140 

89232 

10 

51 

9.654808 

41.6 

9  950458 

10.7 

9.704350 

52.2 

10-295650 

45166 

89219 

9 

62 

655058 

41.6 

950394 

10.7 

704663 

52.2 

295337 

45192 

89206 

8 

53 
54 

655307 
655556 

41.6 
41.5 

950330 
950366 

10.7 
10.7 

704977 
705290 

52.2 
52.2 

296023 
294710 

4521889193 
4524889180 

7 
6 

55 

655805 

41.5 

950202 

10.7 

705603 

52.2 

294397 

4526989167 

5 

56 

656054 

41.6 

950138 

10.7 

705916 

52.1 

294084 

45295 

89153 

4 

57 

656302 

41.4 

950074 

10.7 

706228 

52.1 

293772 

45321 

89140 

3 

58 

656551 

41.4 

950010 

10.7 

706641 

-0   * 

293459 

45347 

89127 

2 

69 

656799 

41.4 

949946 

10.7 

706854 

y2  .  1 

293146 

45373 

89114 

1 

60 

657047 

41.3 

949881 

10.7 

707166 

52.1 

292834 

46399 

89101 

0 

Cosine. 

Sine. 

Cctang  i 

Tang. 

JS.  cos. 

N.sine. 

~r 

63  Degrees. 

48          Log.  Sines  and  Tangents.  (27°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  10" 

Cosine. 

D.  1U" 

Taug. 

D.  10" 

Ootaiig.  i  N.  sine. 

N.  cos. 

0 

9.657047 

9.949881 

9.707166 

10.  292834!  145399 

89101 

60 

1 

667295 

41.3 

A  1  Q 

949816 

10.7 

1  A  T 

707478 

52.0 

KO  (\ 

292522  45425 

89087 

59 

2 

657642 

41  .0 

949752 

1U.  / 

707790 

OZ  .  \) 
KO  A 

292210 

45451 

89074 

58 

3 

657790 

41.2 

A  1  O 

949688 

10.7 

I  A  Q 

708102 

o2.0 

KO  A 

291898 

45477 

89061 

57 

4 

658037 

41  .Z 
A1  O 

949623 

1U.  o 
1  O  ft 

708414 

oz  .  u 

fil  Q 

291586!  45603 

89048 

56 

5 

658284 

41  .Z 
Al  O 

949558 

1  U.  o 
i  A-C 

708726 

01  .  y 
Mo 

291274  45529 

89035 

55 

6 

658531 

41  .  Z 
41  1 

949494 

1U  .  o 
1  0  ft 

709037 

.y 

fi1  Q 

290963  45554 

89021 

54 

7 

658778 

41  .  1 

A-l   I 

949429 

1U«  o 
1  0  ft 

709349 

01  .  y 

K1  Q 

290651  45580 

89008 

53 

8 

659025 

41  »  1 
41  1 

949364 

lv  .  O 

i  n  ft 

709660 

01  .  y 

K1  Q 

290340]  45606 

88995 

52 

9 

659271 

41  .  1 
41  0 

949300 

1U  .  o 
1  0  ft 

709971 

01  .  y 

MQ 

290029  45632 

88981 

51 

10 

659517 

41  .  U 
41  0 

949235 

lu  .  o 
108 

710282 

•  o 

C1   0 

289718  !  45658 

88968 

50 

11 

9.659763 

41  .  V 

41  0 

9.949170 

iu»  o 

10  ft 

9.710693 

Ol  .0 
K1  8 

10.289407 

45684 

88955 

49 

12 

660009 

41  .  \J 
40  Q 

949105 

1U  .  o 
10  ft 

710904 

Ol  .0 

C1   0 

289096 

45710 

88942 

48 

13 

660255 

4i/  .  y 

Af\  n 

949040 

1U  .0 

1  A  Q 

711215 

Ol  .  o 

K1   Q 

288785  146736 

88928 

47 

14 

660501 

4U.y 

40  Q 

948976 

lU.o 
10  ft 

711525 

Ol  .0 

ci  7 

288475  1(45762 

88915 

46 

15 

660746 

4v  .  y 

40  Q 

948910 

Iv.O 

10  ft 

711836 

Ol  .  / 

K-1   17 

288164 

45787 

88902 

45 

16 

660991 

4u.  y 

40  ft 

948845 

1U  .0 
10  ft 

712146 

01  .  / 
c-i  rt 

287854 

45813 

88888 

44 

17 

18 

661236 
661481 

4U  .  o 

40.8 

Af\  Q 

948780 
948715 

1U.O 

10.9 

1  f\  n 

712456 
712766 

01  .  / 

51.7 

287544  i  45839 
287234  145865 

88875 
88862 

43 

42 

19 

20 
21 

661726 
661970 
9.662214 

4U.o 

40.7 
40.7 
40  7 

948650 
948584 
9.948519 

iu.y 
10.9 
10.9 

1O  Q 

713076 
713386 
9.713696 

51  .6 
51.6 
51.6 

K1  fi 

286924||45891 
286614  45917 
10.286304  1  45942 

88848 
88835 
88822 

41 
40 
39 

22 

662459 

4U  .  / 

948464 

iu.  y 

714005 

O  1  .  0 

285995 

45968 

888  OS 

38 

23 

662703 

40.7 

948388 

10.9 

714314 

51  .6 

285686 

45994 

88795 

37 

24 

662946 

40.6 
40  fi 

948323 

10.9 

1  O  Q 

714624 

51  .5 

KI  e 

285376 

46020 

88782 

36 

25 

663190 

4v  .D 

948257 

iu.  y 

714933 

0  1  .  D 

285067 

46046 

88768 

35 

26 

663433 

40.6 

948192 

10.9 

715242 

51  .5 

284758 

46072 

88755 

34 

27 

663677 

40.6 

948126 

10.9 

715551 

51.5 

284449 

46097 

88741 

33 

28 

663920 

40.5 

948060 

10.9 

715860 

51.4 

284140 

46123 

88728 

32 

29 

664163 

40.5 

947995 

10.9 

716168 

51.4 

283832 

146149 

88715 

31 

30 

664406 

40.5 

947929 

11.0 

-It   A 

716477 

51  .4 

283523 

146175 

88701 

30 

31 

9.664648 

40.4 
40  4 

9.947863 

11.0 

no 

9.716785  2J-* 

10.283215 

1  46201 

88U88 

29 

32 

664891 

*±17  .  *± 

40  4 

947797 

.  u 

nA 

717093 

U  -L  .** 

K1  Q 

282907 

46226 

88674 

28 

33 

665133 

4v  .  TC 

40  ^ 

947731 

.  \J 
nA 

717401 

Dl  .  u 

K1   O 

282599 

46252 

88661 

27 

34 

665376 

4U  .  o 

40  3 

947665 

.  U 

nA 

717709  ^-° 

282291 

46278 

88U47 

26 

35 

665617 

4U.  o 

40  ^ 

947600 

•  U 

UA 

718017 

UJ,  .  U 

K1   O 

281983 

46304 

88634 

25 

36 

665859 

4v  .  O 

40  1 

947533 

•  U 

UA 

718325 

Ol  .  O 

K.1  Q 

281675 

46330 

886'20 

24 

37 

666100 

4U  .  Z 

40  1 

947467 

.U 

UA 

718633 

Ol  .  O 

K1   O 

281367 

46355 

8b60? 

23 

38 

666342 

4v  .  Z 

40  <2 

947401 

.  U 

UA 

718940  £•? 

281060 

46381 

38593 

22 

39 

666583 

4v  .  Z 

40  2 

947335 

.  U 

nA 

719248  g-J 

280752 

46407 

88580 

21 

40 

666824 

rtv/.  Z 

A(\   •< 

947269 

.  U 
n{\ 

719565 

K1   O 

280445 

46433 

88666 

20 

41 

9.667065 

4U.  1 
40  1 

9.947203 

.  u 

nA 

9.719862 

51  .Z 

Kl   0 

10.280138 

46458 

88553 

19 

42 

667305 

ftU  .  1 

A  A  1 

947136 

.  U 

720169  £•:? 

279831 

46484 

88639 

18 

43 

667546 

4U.  1 

A  A  1 

947070 

11.1 

730*76  51'? 

279524 

46510 

88526 

17 

44 

667786 

4U.  1 

947004 

11.1 

720783 

Ol  .1 

279217 

46536 

88512 

16 

46 
46 

668027 
668267 

40.0 
40  0 

946937 
946871 

11  .1 
11.1 

721089 
721396 

51.1 

51.1 

278911 
278604 

46561 

1  46587 

88499 
88485 

15 
14 

47 

668506 

40.0 

QQ  Q 

946804 

11  .  1 
U1 

721702  jjJ-J 

278298 

46613 

88472 

13 

48 

668746 

oy  .  y 
00  q 

946738 

.  1 
n| 

722009  f'Q 

277991 

1  46639 

88458 

1? 

49 

668986 

oy  .  y 
39  9 

946671 

.  1 

HI 

722315!  g.  0 

277(-;85 

46664 

88445 

n 

50 

669225 

QQ  Q 

946604 

f  1 
nl 

722621  °}'X 

277379 

46690 

884ol 

10 

61 

9.669464 

oy  .y 

QQ  G 

9.946538 

.  1 
n-i 

9.722927  |«-g 

1?  277073 

46716 

88417 

9 

52 

669703 

oy  .0 

on  u 

946471 

.  1 
HI 

733232  6J.O 

276768 

46742 

88404 

8 

53 

669942 

oy  .  o 

946404 

.  .  1 

723638  ^-^ 

276462 

46767 

88390 

7 

54 

670181 

39.8 

QQ  T 

946337 

11.1 
nl 

723844  ™-9 

276156 

1  46793 

8837? 

6 

65 

670419 

oy  .  / 

946270 

.  1 

724149  igJJ'J 

.275851 

46819 

88363 

5 

56 

670658 

39.7 

946203 

11.2 

724454  ^'9 

276546 

46844 

88349 

4 

57 

670896 

39.7 

946136 

11.2 

724759  SI  » 

276241 

46870 

88336 

3 

58 

671134 

39  .7 

946069 

11  .2 

738066  JJ-g 

274935 

46896 

88322 

2 

59 

671372 

39  .6 

QQ  fi 

946002 

11.2 

no 

725369  ,°"-« 

274631 

46921 

88308 

1 

60 

671609 

oy  .  D 

945935 

.Z 

725674  6U'b 

274326 

46947 

88295 

0 

CoHine. 

Sine. 

Cotang. 

Tang 

N.  CO8. 

A.  Him-. 

~~7 

62  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (28°)  Natural  Sines.          49 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  1(K 

Cotang.   N.  sine 

N.  cos 

0 

9.671609 

9.945935 

9.725674 

10.274326  46947 

88295 

60 

1 

671847 

39.6 

945868 

11.2 

725979 

50.8 

274021  146973 

88281 

59 

2 

672084 

39.5 

945800 

11  .2 

726284 

50.8 

273716  146999 

88267 

58 

3 

672321 

39.5 

945733 

11.2 

726588 

50.7 

2734121147024 

88254 

57 

4 

672558 

39.5 

OQ   K 

945666 

11.2 
net 

726892 

50.7 

Kft  7 

273108 

47050 

88240 

56 

6 

672795 

Oy  .  O 

qq  A 

945598 

.  z 

no 

727197 

ou  .  / 

KA  rj 

272803 

47076 

88226 

56 

6 

673032 

Oy  .  *± 
OQ   A 

945531 

•  Z 
no 

727501 

OU.  t 

272499 

147101 

88213 

54 

7 

673268 

e>y  .4 

945464 

.Z 

727805 

60-7 

272195 

i  47127 

88199 

53 

8 

673505 

39.4 

on  A 

945396 

11.3 

no 

728109 

50.6 

KA  £J 

271891 

i  47153 

88185 

52 

9 

673741 

oy  .  4 

945328 

.0 

728412 

ou.o 

271588 

47178 

88172 

51 

10 

673977 

39.3 

945261 

11.3 

728716 

50.6 

271284 

47204 

88158 

50 

11 

9.674213 

39.3 

9.945193 

11.3 

9.729020 

50.6 

10.270980 

47229 

88144 

49 

12 

674448 

39.3 

945125 

11  .3 

729323 

50-6 

270677 

47255 

88130 

48 

13 

674684 

39.2 

on  q 

945058 

11.3 

nq 

729626 

50.5 

Kf\   K 

270374 

47281 

88117 

47 

14 

674919 

oy  ,  •& 
on  o 

944990 

•  O 
no 

729929 

o().o 

270071 

47306 

88103 

46 

15 

675155 

oy  .  Ji 

944922 

.0 

730233 

50-5 

269767 

47332 

88089 

45 

16 

675390 

39.2 

944854 

11.3 

730535 

5Q.5 

269465 

47358 

88075 

44 

17 

675624 

39.1 

944786 

11.3 

730838 

50.5 

269162 

47383 

88062 

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30 
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265537  4769087896 

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943624 
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11.5 
11.5 
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735969 
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264031 
263731 
263430 

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47844 
47869 

87826 
87812 
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26 
25 
24 

37 

680288 

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943417 

11  .  5 

nK 

736871 

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263129  i  47895 

87784 

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680519 

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943348 

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737171 

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262829 

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943279 

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943210 

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9.943141 

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47997 

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943072 

11  .6 

738371 

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261629 

48022 

87715 

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43 
44 

681674 
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943003 
942934 

11.5 
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738671 
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261329 
261029 

48048 
48073 

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87687 

17 
16 

45 

682135 

00.4 

942864 

11.6 

739271 

49.9 

260729  !  48099 

87673 

15 

46 

682366 

38.4 

942795 

11.6 

739570 

49.9 

260430  48124 

87659 

14 

47 
48 

682595 
682825 

38.  3 
38.3 

qQ   q 

942726 
942656 

11.6 
11.6 

nc 

739870 
740169 

49.9 
49.9 

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2601301  48150 
2598311  48175 

87645 
87631 

13 
12 

49 

683055 

OO  .  O 
qQ  q 

942587 

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nil 

740468 

4y  .y 

4.Q  K 

259532  !  48201 

87617 

11 

50 

683284 

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oQ  o 

942517 

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n/J 

740767 

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AQ  8 

259233  48226 

87603 

10 

51 
52 
53 

9.683514 
683743 
683972 

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38.2 
38.2 

9.942448 
942378 
942308 

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11.6 
11.6 

9.741066 
741365 
741664 

4y  .0 
49.8 
49.8 

Af\  Q 

10.2589341  48252 
258636  48277 
2583361  48303 

87589 
87575 
87561 

9 

8 

7 

54 

684201 

38.2 

qQ   1 

942239 

11.6 
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741962 

4y  .0 

_4Q  rr 

258038  48328 

87546 

6 

55 

684430 

OO  .  1 

942169 

.0 

742261 

•iy  .  / 

257739  J48354 

87532 

6 

56 
57 

684658 
684887 

38.1 
38.1 

942099 
942029 

11.6 
11.6 

742559 

742858 

49.7 
49.7 

257441  48379 
257142  48405 

87518 
87604 

4 
3 

58 

685115 

38.0 

941959 

11.6 

743156 

49.7 

256844 

48430 

87490 

2 

59 

685343 

38.0 

941889 

11.6 

743454 

49.7 

256546 

48456 

87476 

1 

60 

685671 

38.0 

941819 

11.7 

743752 

49.7 

266248 

48481 

87462 

0 

Cosine. 

Sine. 

Cotang. 

Tamv   N7co7. 

N.sine. 

i 

61  Degrees. 

50          Log.  Sines  and  Tangents.  (29°)  Natural  Sines.     TABLE  II. 

Sine. 

D.  10" 

Cosine.  |D.  10" 

Tang. 

D.  10" 

Cotang. 

N.  sine 

N.  cos. 

0 

9.685571 

00  n 

9.941819 

nry 

9.743762 

Af\   f* 

10.256248 

48481 

87462 

60 

1 

685799 

oo  ,  (j 

07  Q 

941749 

.  / 

nrt 

744050 

4y  .D 

AQ  fi 

265950 

48506 

87448 

69 

2 

686027 

o  /  ,  y 
07  q 

941679 

.  / 
U7 

744348 

4y  •  o 

AQ  fi 

265652 

48532 

87434 

68 

3 

686254 

o  /  .  y 

07  q 

941609 

.  / 
U7 

744646 

4y  •  o 

AQ  ft 

255355 

48557 

87420 

67 

!  4 

686482 

o  /  .  y 

0-7  Q 

941539 

.  i 

Uij 

744943 

^y  *  o 

AQ  ft 

255057 

48583 

87406 

66 

5 

686709 

o  /  .  y 

07  Q 

941469 

.  / 

nrr 

745240 

4y  »  o 

A.Q  fi 

264760 

48608 

87391 

65 

6 

686936 

O  I  ,  O 
07  Q 

941398 

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U7 

745638 

4y  .  o 

AQ  fi 

254462 

48634 

87377 

54 

7 

687163 

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07  Q 

941328 

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745835 

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AQ  fi 

254165 

48659 

87363 

53 

8 

687389 

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941258 

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746132 

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253868 

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87349 

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941187 

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746429 

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253571 

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87335 

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10 

687843 

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941117 

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746726 

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253274 

i  48735 

87321 

60 

11 

9.688069 

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07  17 

9.941046 

.  / 

Uo 

9.747023 

4y  .0 

AQ  A 

10.252977 

i  48761 

87306 

49 

12 

688295 

O  1  .  1 

07  7 

940975 

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no 

747319 

4y  .  TC 

AQ  A 

252681 

48786 

87292 

48 

13 

688521 

O  /  .  / 

07  ft 

940905 

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no 

747616 

4y  .4 

AQ  A 

252384 

48811 

87278 

47 

14 
15 

688747 
688972 

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37.6 

07  fi 

940834 
940763 

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11.8 

no 

747913 

748209 

4y  .  TC 
49.4 

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252087 
251791 

48837 
48862 

87264 
87260 

46 
46 

16 

689198 

o  /  .  D 

07  ft 

940693 

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748505 

4y.4 

AQ  Q 

261495 

148888 

87235 

44 

17 

689423 

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940622 

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748801 

rri?  .  O 

4.Q  ^ 

251199  148913 

87221 

43 

18 

689648 

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07  X 

940551 

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749097 

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250903 

i  48938 

87207 

42 

19 

689873 

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07  K 

940480 

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749393 

rty  .  O 

49  3 

260607 

148964 

87193 

41 

20 

690098 

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940409 

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749689 

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250311 

14898987178 

40 

21 

9.690323 

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9.940338 

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9.749986 

rt^  .  o 
AQ  ^l 

10.250015 

4901487164 

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690548 
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940267 
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49.2 

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249719 
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49065  87136 

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690996 
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940125 
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248538 

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87093 

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939911 
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11.9 
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751767 
752052 

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49.2 
49 

248243 
247948 

49166 
49192 

87079 

87064 

33 
32 

29 

692116 

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939768 

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752347 

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247653  49217 

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762642 

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247358  149242 

87036 

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9.752937 

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10.247063  149268 

87021 

29 

32 
33 
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692785 
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37.1 
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939554 
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753231 
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246769  [49293 
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754703 

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245297  49419 

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764997 

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49  0 

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86921 

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765291 

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49  0 

2447091  149470 

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765685 

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2444161  49495 

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10.244122  |49521 

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42 
43 
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695007 
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938836 
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12.0 
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756172 
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243828 
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149546 
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86863 
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938619 

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120 

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46 
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36.8 

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938647 
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12.0 

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757346 

757638 

T:O  .  y 
48.8 
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242656 
242362 

49647 
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86805 
86791 

14 
13 

48 

696334 

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938402 

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757931 

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242069 

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86777 

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50 

696554 
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938330 
938268 

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12 

768224 
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48.8 
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52 

9.696995 
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9.938185 
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9.758810 
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10.241190  4977386733 
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8 

53 

697435- 

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938040 

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19, 

769395 

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48  7 

240605 

4982486704 

7 

54 

697654 

OO.D 
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937967 

»*». 

12 

759687 

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240313 

49849 

86690 

6 

56 

697874 

OO  .D 

Oft  £» 

937895 

12 

769979 

r4O  .  f 

48  7 

240021 

49874 

86675 

6 

56 

698094 

OO.O 

Oft  C 

937822 

12* 

760272 

^tO  .  / 

48  7 

239728 

49899 

86661 

4 

57 

698313 

uO  .  D 

qft  C 

937749 

12 

760564 

48*7 

239436 

49924 

86646 

3 

58 

698532 

oO  .  O 

c\r»  e 

937676 

1  O 

760856 

AQ  « 

239144 

49950 

86632 

2 

59 

698751 

OO.O 
qft  e 

937604 

1  -  • 
12. 

761148 

rrO  .  O 

48.6 

238852 

49976 

86617 

1 

60 

698970 

oO  .  O 

937631 

761439 

238561 

50000 

86603 

0 

Cosine. 

Sine. 

Cotang. 

Tang.   I  N.  co«. 

iViMlH'. 

""^ 

60  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (30°)  Natural  Sines.          51 

' 

Sine.  ;D.  10' 

Cosine. 

D.  10' 

Tan>. 

D.  10 

Cotang.   N.  sine 

N.  cos 

0 

9.698970 

Oft  A 

9.937531 

1  O  i 

9.761439 

48  6 

10.238561  50000 

86603 

60 

1 

699189 

OO  .  *T 

Oft  A 

937458 

1  Z  .  1 
199 

761731 

Aft  e. 

238269 

50025 

86588 

59 

2 

699407 

OO  .  T: 

937385 

iZ  .  Z 

762023 

••to  .  D 

237977 

50050 

86573 

58 

2 

699626  ™'* 

937312 

12.2 

1  O  O 

762314 

48.6 

AQ  fc 

237686  1  60076 

86559 

57 

4 

699844  *•* 

937238 

iZ  .  Z 

762608 

4o  .b 

237394 

50101 

86544 

56 

6 
6 

700062  XT, 
700-280  !^-J 

937165 
937092 

12.2 
12.2 

762897 
763188 

48.5 
48.5 

237103 
236812 

50126 
50151 

86530 
86515 

55 
54 

7 

700498  rJ'o 

937019 

12.2 

199 

763479 

48.6 
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236521  50176 

86501 

53 

8 

700716  ™-, 

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199 

763770 

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48  F 

236230 

50201 

86486 

52 

9 

700933  ^'^ 

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764061 

235939 

50227 

86471 

51 

10 

701151  !*'„ 

936799 

12.2 

764352 

48.5 

235648  150252 

86457 

50 

11 

12 

9.701368 
701585 

oo.  z 
36.2 

9.936725 
936652 

12.2 
12.2 

9.764643 
764933 

48.4 
48.4 

10.235357 
235087 

|  50277 
50302 

86442 
86427 

49 

48 

13 

701802 

36.2 

936578 

12.3 

765224 

48.4 

234776 

60327 

86413 

47 

14 

702019 

36.  1 

Of-   1 

936505 

12.3 

765514 

48.4 

234486 

50352 

86398 

46 

15 

702236 

oo.  1 

Oft  1 

936431 

12.3 

1O  Q 

765805 

48.4 

234195 

50377 

86384 

45 

16 

702452 

OD  •  1 

936357 

IZ  .  0 

766095 

4o.4 

233905 

50403 

86369 

44 

17 

702669 

36.1 

936284 

12.3 

766385 

48.4 

233615 

50428 

86354 

43 

18 

702885 

36  .  0 

936210 

12.3 

766675 

48.3 

233325 

50453 

86340 

42 

19 

703101 

36.0 

936136 

12.3 

766965 

48.3 

233035 

50478 

86325 

41 

20 

703317 

36.0 

936062 

12.3 

767255 

48.3 

232745 

60503 

86310 

40 

21 

9.703533 

36.  0 

9.936988 

12.3 

9.767545 

48.3 

10.232455 

50528 

86295 

39 

22 

703749 

35.9 

Q"  O 

935914 

12.3 

767834 

48.3 

232166 

60553 

86281 

38 

23 

703964 

oo.y 

3-  Q 

935840 

12.3 

768124 

48.3 

231876 

60578 

86266 

37 

24 

704179 

o  .  y 

935766 

12.3 

768413 

48.  2 

231587 

60603 

86251 

36 

25 

704395 

o-°Q 

935692 

12.4 

1O  A 

768703 

48.2 

231297 

50628 

86237 

35 

26 

704610 

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935618 

12.4 

768992 

48.2 

231008 

50654 

86222 

34 

27 
28 

704825 
705040 

oo.o 
35.8 

Q1"  Q 

935543 
935469 

12.4 
12.4 

769281 
769570 

48.2 
48.2 

230719 
230430 

50679 
50704 

86207 
86192 

33 
32 

29 

705254 

oo.o 

Q£  Q 

935395 

12.4 

769860 

48.2 

230140 

50729 

86178 

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30 

705469 

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935320 

12.4 

770148 

48.1 

229852 

50754 

86163 

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31 

9.705683 

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Q~  T 

9.935246 

12.4 

9.770437 

48.1 

10-229563 

60779 

86148 

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32 

706898 

00.  / 

o"  n 

935171 

12.4 

770726 

48.1 

229274 

50804 

86133 

28 

33 

706112 

oo.  i 

3"  n 

935097 

12.4 

771016 

48.1 

228986 

50829 

86119 

27 

34 

706326 

O.  I 

935022 

12.4 

771303 

48.1 

228697 

50854 

86104 

26 

35 

706539 

35.6 

934948 

12.4 

771592 

48.1 

228408 

50879 

86089 

25 

36 

706753 

35.6 

934873 

12.4 

771880 

48.1 

228120 

50904 

86074 

24 

37 

706967 

35.6 

934798 

12.4 

772168 

48.0 

227832 

50929 

86059 

23 

38 

707180 

35.6 

934723 

12.6 

772457 

48.0 

227543 

50954 

86045 

22 

39 
40 

707393 
707606 

35.5 
35.5 

934649 
934574 

12.5 
12.6 

772746 
773033 

48.0 
48.0 

227255 
226967 

50979 
51004 

86030 
86015 

21 
20 

41 

9.707819 

35.6 

9-934499 

12.5 

9.773321 

48.0 

10-226679 

51029 

86000 

19 

42 

708032 

35.6 

934424 

12.5 

773608 

48.0 

226392 

51054 

85985 

18 

43 

708245 

35.4 

934349 

12.5 

773896 

47.9 

226104 

61079 

85970 

17 

44 

708458 

35.4 

934274 

12.5 

774184 

47.9 

225816 

61104 

85956 

16 

45 

703670 

35.4 

934199 

12.5 

774471 

47.9 

225529 

51129 

85941 

15 

46 

708882 

35.4 

934123 

12.5 

774759 

47.9 

225241 

51154 

85926 

14 

47 

709094 

35.3 

934048 

12.5 

775046 

47.9 

224954 

51179 

85911 

13 

48 

709303 

35.3 

933973 

12.5 

776333 

47.9 

224667 

51204 

85896 

12 

49 

709518 

35.3 

933898 

12.5 

775621 

47.9 

224379 

61229 

85881 

11 

50 

709730 

35.3 

933822 

t2.6 

775908 

47.8 

224092 

61254 

85866 

10 

51 

9.709941 

35.3 

9.933747 

12.6 

9.776195 

47.8 

10-223805 

51279 

85851 

9 

52 

710153 

35.2 

933671 

12.6 

776482 

47.8 

223518 

51304 

85836 

8 

53 

710364 

35.2 

933596 

12.6 

776769 

47.8 

223231 

61329 

85821 

7 

64 

710575 

35.2 

933520 

12.6 

777056 

47.8 

A1  Q 

222945 

51354 

85806 

6 

55 

710786 

35.2 

933445 

12.6 

777342 

47.O 

222658 

51379 

86792 

5 

66 

710967 

35.  1 

933369 

12.6 

777628 

47.8 

222372 

51404 

85777 

4 

57 

711208 

35.1 

933293 

12.6 

777915 

47.7 

222085 

51429 

85762 

3 

68 

711419 

35.1 

933217 

12.6 

778201 

47.7 

221799 

61454 

35747 

2 

59 

711629 

35.1 

933141 

12.6 

778487 

47.7 

221612 

51479 

35732 

60 

711839 

35.0 

933066 

12.6 

778774 

47.7 

221226 

51504 

35717 

0 

Cosine. 

Sine 

Co  tang. 

Tang.  . 

N.  co*. 

N.siue. 

' 

59  Degrees. 

52          Log.  Sines  and  Tangents.  (31°)  Natural  Sines.     TABLE  II. 

' 

Sine.   |D.  10" 

Cosine. 

D.  10" 

Taug. 

Dv  10" 

Cotang. 

N.sine.lN.  cos.- 

0 

9.711839 

3.933036 

9.778774 

An  n 

10.221226 

51504|S5717 

60 

1 

712050 

35.0 

932990 

12.6 

779080 

47  .7 

220940 

51629(86702 

59 

2 

712260 

35.0 

932914 

12.7 

779346 

A7  fi 

220654 

51554185687 

58 

3 

712469 

35.0 

Q/i   Q 

93283S 

12.  7 

19  7 

779832 

4,7  fi 

220368 

5157985672 

57 

4 

712679 

d4.y 

932762 

i  -<  .  i 

779918 

*i/  .O 

An  K. 

220082 

51tf04|85657 

56 

5 

712889 

34.9 

932885 

10  n 

780203 

4<  .O 

219797 

5162885642 

55 

6 

713098 

34.9 

932609 

!;«.  7 
10  n 

780489 

A7  fi 

219511 

51653 

85627 

54 

7 
8 

713308 
713517 

34.9 
34.9 

932533 
932457 

I*  .  7 

12.7 

780775 
781060 

47^6 

219225 
218940 

51678 
61703 

85612 
85597 

53 
52 

9 

713726 

34.8 

O  i  Q 

932380 

12.7 

781346 

47.6 

218654  51728}855«2 

61 

10 

713935 

d4.O 
O/i  Q 

932304 

10  rf 

781631 

An  F» 

218369!  6175385567 

60 

11 

9.714144 

d4.O 

9.932228 

1-4  .  7 

9.781916 

4/  .  0 

10.  218084  '  51778185551 

49 

12 
13 

714352 
714561 

34.8 
34.7 
o/i  n 

932151 
932075 

12.7 
12.7 

1O  Q 

782201 
782486 

47.5 

47.5 

4.7  Ft 

217799  51803185536 

217514  i  61828  85521 

48 

47 

14 

714769 

d4.  / 

931998 

Ixi.O 

782771 

4/  .  D 

Act  K 

217229 

61852 

85506 

46 

15 

714978 

34.7 

n  A  ri 

931921 

12.8 

783058 

47  .6 
4.7  Ft 

216944)  51877 

85491 

45 

16 

715186 

d4.  / 

o/i  rr 

931845 

1O  Q 

783341 

Tb  /  .O 

216659  |61902|86476 

44 

17 

18 
19 
20 
21 

22 
23 

715394 
715602 
715809 
716017 
9.716224 
716432 
716639 

d4.  / 

34.6 
34.6 
34.6 
34.6 
34.5 
34.6 

OX  K 

931768 
931691 
931614 
931637 
9.931460 
931383 
931306 

11S.O 

12.8 
12.8 
12.8 
12.8 
12.8 
12.8 

1O  Q 

783626 
783910 
784195 
784479 
9.784764 
785048 
785332 

47.4 
47.4 
47.4 

47.4 
47.4 
47.4 

216374 
216090 
215805 
215521 
10.215236 
214962 
214668 

51927185461 
61952185446 
51977185431 
i  52002185416 
62026J854U1 
1  62051  85385 
52076|85370 

43 
42 
41 
40 
39 
38 
37 

24 

716846 

d4.O 

931229 

1-6.0 

785616 

47.3 

214384(162101 

85365 

36 

26 
26 

717053 
717259 

34.5 
34.5 

OX  A 

931152 
931075 

12.9 
12.9 

785900 
786184 

47.3 
47.3 

214100  i  152126 
213816  52161 

86340 

85325 

36 

34 

27 

717466 

d4.4 

930998 

12.9 

786468 

47.  d 

213632  62175 

85310 

33 

28 
29 

717673 

717879 

34.4 
34.4 

OX  A 

930921 
930843 

12.9 
12.9 

786752 
787036 

47.3 
47.3 

213248 
212964 

!  52200 
52226 

85294 
85279 

32 
31 

30 

718086 

d4.4 

930766 

12.9 

787319 

47.3 

212681 

52250852u4 

30 

31 

9.718291 

34.3 

OX  Q 

9.930688 

12.9 

9.787603 

47.2 

An  o 

10.212397 

62275  85249  1  29 

32 
33 

718497 
718703 

d4.d 

34.3 

OX  Q 

930611 
930533 

12.  9 
12.9 

787886 
788170 

4/  .  & 

47.2 

212114 
211830 

52299 

62324 

85234 
85218 

28 
27 

34 

718909 

d4.d 

930456 

12.9 

788453 

47.2 

211647 

1  52349 

86203 

26 

35 

719114 

34.3 

930378 

12.9 

788736 

47.2 

211264 

52374 

85188 

25 

36 

719320 

34.2 

930300 

12.9 

789019 

7!'? 

210981 

52399 

85173 

24 

37 

719526 

34.2 

930223 

13.0 

789302 

47  .2 
An  1 

210tiy8 

52423 

85167 

23 

38 
39 

719730 
719935 

34.2 
34.2 

OX   1 

930145 
930067 

13.0 
13.0 

789585 
789868 

4<  .  1 

47.1 

An  i 

210416 
210132 

62448 

52473 

85142 
85127 

22 
21 

40 

720140 

d4.1 

929989 

13.0 

790151 

4/  .  1 

209849 

52498 

85112 

20 

41 

9.720345 

34.  1 

01    "I 

9.929911 

13.0 

9.790433 

A7 

10.209567 

52522 

85096 

19 

42 
43 

720549  j  34  'J 
720754  !*H 

929833 
929755 

13.0 
13.0 

790716 
790999 

47! 

An 

209284 
209001 

62647 
52572 

85081 
85066 

18 
17 

44 

720958  jij;  'n 

929677 

13.0 

791281 

4/  . 

208719 

52597 

85051 

16 

46 

721162  £'X 

929599 

13.0 

791563 

47. 

208437 

52621 

850^5 

15 

46 

721366  1  XT  n 

929521 

13.0 

791846 

47.0 
An  f\ 

208154 

62646 

851)20 

14 

47 

731670  JS' 

929442 

13.0 

792128 

4/  .U 

207872 

52671 

85005 

IS 

48 

731774  JJ  a 

929364 

13.0 

1  Q  1 

792410 

4.7  0 

207690 

62696 

84989 

12 

49 

721978  JJ'Si 

929286 

Id.  1 

792692 

4l  .  U 

207308 

62720 

84974 

11 

60 

722181  J;'q 

929207 

13.1 

792974 

47  »0 

207026 

52745 

84959 

10 

51 

9.722385  JJq 

9.929129 

13.1 

9.793256 

7 

10.206744 

52770 

84943 

9 

52 

722588  ~{Q 

929050 

13.1 

793538 

Af*   l\ 

206462 

52794 

84928 

8 

53 

722791  J;'Q 

928972 

13.  1 

793819 

4o.y 

206181 

62819 

84913 

7 

64 

722994  ^'fl 

928893 

13.1 

794101 

46  .  9 

205899 

52844 

84897 

6 

65 

723197  ^o'o 

928815 

13.1 

1  Q  1 

794383 

46  .9 
4.fi  Q 

205617 

62869 

84882 

5 

56 

723400  ^o'o 

928736 

Id.  1 

794664 

40  .  y 

X£l  O 

205336 

52893 

84866 

4 

67 

723603  JJ'r 

928667 

13.1 

794946 

4o.y 

205055 

52918 

84851 

3 

68 

723805',  XT, 

928578 

13.1 

796227 

4o.y 

204773 

62943 

84836 

2 

69 

724007  £r' 

928499 

13.1 

795508 

46.9 

204492 

52967 

84820 

1 

60 

724210  , 

928420 

13.1 

795789 

46.8 

204211 

62992 

84805 

0 

Cosine. 

Sine. 

Cotang. 

Tang.   '  N.  cos. 

N.~8iue~. 

~ 

58  Degrees. 

TABLE  TI.     Log.  Sines  and  Tangents.  (32°)  Natural  Sines. 

53 

i 

Sine. 

D.  10" 

Cosine.  |D.  10" 

Tffpg.  ' 

D.  10"|  Cotang.   N.  siue. 

N.  cos. 

0 

9.724210 

9.928420 

9.795789 

A,,  R  10.204211 

!  52992 

84805 

60 

1 

724412 

33.7 

928342 

13.2 

796070 

4b.o 

AC   Q 

203D30 

153017 

84789 

59 

2 

724614 

33.7 

928263 

13.2 

796351 

4o.o 

203649 

!  63041 

84774 

68 

3 

724816 

33.6 

928183 

13.2 

796632 

46.8 

Ad   O 

203368 

i  63086 

84759 

57 

4 

725017 

33.6 

928104 

13.2 

796913 

4b.o 

AC   O 

203087 

63091 

84743 

56 

5 

725219 

33.6 

928025 

13.2 

797194 

4o.o 

AC  Q 

202806 

53115 

84728 

55 

6 

725420 

33.6 

927946 

13.2 

797475 

4o.o 

AC  Q 

202625 

63140 

84712 

54 

7 

725622 

33.5 

927867 

13.2 

797756 

45.0 

AC  Q 

202246 

53164 

84697 

53 

8 

725823 

33.5 

927787 

13.2 

798036 

4o.o 

2019641163189 

84681 

52 

9 

726024 

33.5 

927708 

13.2 

798316 

46.7 

201684 

53214 

84666 

61 

10 

726225 

33.5 

927629 

13.2 

798596 

46.7 

201404 

53238 

84660 

50 

11 

9.726426 

33.5 

qo  A 

9.927549 

13.2 

•too 

9.798877 

46.7 

AC.   7 

10.201123 

53263 

84635 

49 

12 

726626 

OO  .  *x 
OO  A 

927470 

I  O  .  Z 

i  q  o 

799167 

40  .  1 

AC  7 

200843 

53288|84619 

48 

13 

726827 

OO  ,  rt 

927390 

lo  .  o 

799437 

40  .  1 

200563 

63312 

84604 

47 

14 

727027 

33.4 

oo  A 

927310 

13.3 

100 

799717 

46.7 

Af\  7 

200283 

5333784588 

46 

15 

727228 

OO  .  Q 

927231 

lo  .  o 
10  o 

799997 

4O.  / 

AC   C 

200003 

53361|84573 

45 

16 

727428 

33  .4 

927151 

lo.  o 
10  o 

800277 

4o.o 

AC   C 

199723 

63386184557 

44 

17 

727628 

33.  3 

*>O   q 

927071 

lo.d 
-i  o.  q 

800557 

4b  .0 

AC   C 

199443 

53411 

84542 

43 

18 

727828 

OO  .  O 

926991 

lo  .  o 
10  o 

800836 

4O  .  D 

AC.  C. 

199164 

63435 

84526 

42 

19 

728027 

33.3 

OQ   q 

92691  1 

lo.  d 

•1  0   q 

801116 

4b.b 

AC   C 

198884 

63460 

84511 

41 

20 

728227 

oo.o 

926831 

lo.  o 
to  o 

801396 

4O  .D 

AC   £? 

198604 

53484 

84495 

40 

21 

9.728427 

33.3 

9.926751 

lo.d 

1O   O 

9.801676 

4b.b 

Ac  c 

10.198325 

63509 

84480 

39 

22 

728S26 

33.2 

926671 

lo.d 

10  o 

801955 

4o.o 

AC.  C 

198045 

53534 

84464 

38 

23 

728825 

33.2 

926591 

Id.  3 

802234 

4o.b 

AC.   K. 

197766 

53558 

84448 

37 

24 

729024 

33.2 

926511 

13.3 

1O   1 

802513 

4b.o 

AC  K. 

197487 

53583 

84433 

36 

25 

729223 

33.2 

926431 

Id.  4 
10  * 

802792 

4b.o 

197208 

53607 

84417 

35 

26 

729422 

33.1 

926351 

Id.  4 

10  i 

803072 

46.6 

AC  K. 

196928 

63632 

84402 

34 

27 

729621 

33.1 

926270 

lo.4 

10  1 

803351 

4b.o 

196649 

53656 

84386 

33 

28 

729820 

33.1 

926190 

lo.4 
10  1 

803630 

46.6 

196370 

53681  84370 

32 

29 

730018 

33  .  1 

926110 

Id.  4 

10  i 

803908 

46.6 

196092 

63705  84355 

31 

30 

730216 

33.0 

926029 

Id.  4 

1O   ( 

804187 

46.6 

195813 

53730  84339 

30 

31 

9.730415 

33.0 

9.925949 

Id.  4 

1C    i 

9.804466 

46  .5 

10.195534 

5376484324 

29 

32 
33 

730613 
730811 

33.0 
33.0 

925868 
925788 

Id.  4 
13.4 

804746 
805023 

46.4 
46.4 

195255 
194977 

63779  84308 
63804i84292 

28 
27 

34 

731009 

33.0 

925707 

13.4 

805302 

46.4 

194698 

63828  84277 

26 

35 

731206 

32.9 

925626 

13.4 

805580 

46.4 

./-.    A 

194420 

!  638531  84261 

25 

36 
37 

731404 
731602 

32.9 
32.9 

925545 
925465 

13.4 
13.5 

805859 
806137 

4o.4 
46.4 

194141 
193863 

53877^4245 
53902  '84230 

24 
23 

38 

731799 

32.9 

925384 

13.5 

806415 

46.4 

193585 

6392b'J842l4 

22 

39 

731996 

32.9 

or*  Q 

925303 

13.5 

806693 

46.3 

193307 

i  53951  184198 

21 

40 
41 

732193 
9.732390 

32.0 

32.8 

on  Q 

925222 
9.925141 

13.5 
13.5 

1O  K 

806971 
9.807249 

46.3 
46.3 

AC  Q 

193029 
10.192751 

53975:84182 
5400084167 

20 
19 

42 
43 

732587 

732784 

oii.o 
32.8 

on  ft 

925060 
924979 

lo.O 

13.5 

807627 
807805 

4b.  o 
46.3 

AC   O 

192473 
192195 

|64024!84151 
1  54049  84135 

18 
17 

44 

732980 

o2.o 
oo  7 

924897 

13.  5 

I  q  C 

808083 

4b.o 

AC  q 

191917  |i64073!84120 

16 

45 

733177 

'->  w  .  ' 

09  7 

924816 

lo  .  O 
1  Q  K 

808361 

4b.  o 

AC    O 

191639 

!  54097  ;84  104 

15 

46 

733373 

-  )  ~  .  < 
oo  T 

924735 

lo  .  O 
10  c. 

808638 

4b.o 

191362 

54122;84088 

14 

47 

733669 

62,.  i 
on  7 

924654 

Id.b 

10  c. 

808916 

46.2 

191084 

54146  84072 

13 

48 

733765 

Oil.  I 

924572 

lo.o 

809193 

46.2 

190807 

54171!84057 

12 

49 

733961 

32.7 

924491 

13.6 

809471 

46.2 

190529 

\  541  95  84041 

11 

50 

734157 

32.6 

924409 

13.6 

809748 

46.2 

190252 

54220  84026 

10 

51 

9.734353 

32.6 

9.924328 

13.6 

9.810025 

46.2 

10.189975 

54244  84009 

9 

52 

734549 

32.6 

924246 

13.6 

810302 

46.2 

189698  164269 

83994 

8 

63 

734744 

32.6 

924164 

13.6 

810580 

46.2 

189420  1!  64293 

83978 

7 

54 

734939 

32.5 

924083 

13.6 

810857 

46.2 

189143 

15431783962 

6 

55 

735135 

32.5 

924001 

13.6 

811134 

46.2 

188866 

!  54342'83946 

5 

56 
67 

735330 
735526 

32.6 
32.5 

923919 
923837 

13.6 
13.6 

811410 
811687 

46.1 
46.1 

188590  54366J83930 
188313  I54S91  183915 

4 
3 

58 

735719 

32.6 

923765 

13.6 

811964 

46.1 

188036 

54415  83899 

2 

59 
60 

735914 
736109 

32.4 
32.4 

923673 
923591 

13.7 
13.7 

812241 
812517 

46.1 
46.1 

187759 
187483 

54440  83883 
54464J83867 

1 
0 

Cosine. 

Sine. 

Cotang. 

Tang. 

i  N.  cos. 

X.sine. 

' 

57  Degrees. 

54          Log.  Sines  and  Tangents.  (33°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  lu 

Cosine. 

D.  10 

Tang. 

D.  10 

Cotang. 

N.  sine. 

N.  cos 

0 

9.736109 

9.923591 

9.812517 

10.187482 

1  64464 

83867 

60 

1 

736303 

32.4 

923509 

13.1 

812794 

46.] 

187206  i  64488 

83851 

59 

2 

736498 

32.  4 

923427 

13.1 

813070 

46.] 

186930  !i  6451  3 

83835 

58 

3 

736692 

32.4 

923345 

13.1 

813347 

46.] 

186653  i  64537 

83819 

57 

4 

736886 

32.  c 

qo  <: 

923263 

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10  17 

813623 

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83804 

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923181 

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186101 

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83708 

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14 
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922520 
922438 
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816107 
816382 
816668 

45.9 
45.9 
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183893 
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6480583645 
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741316 

oi  ,y 

921357 

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741889 

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55218  83373 

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177846 

55291J83324 
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25 

36 

743033 

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920604 

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822429 

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177671 

5533983292 

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743223 

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822703 

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177297 

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744171 
744361 
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919931 

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824072 
824345 
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45.  o 
45.6 
45.6 

A  "   ^J 

175928 
175655 
176381 

5548483195 
5550983179 
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18 
17 
16 

45 

744739 

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175107 

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825166 

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174834 

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0  1   K 

919677 

14.1 

825439 

45.6 

174561 

5560583115 

13 

48 
49 
60 

745306 

745494 
745683 

ol  .6 
31.4 
J1.4 

ol   A 

919593 
919508 
919424 

14.1 
14. 
14. 

825713 
825986 
826259 

45.6 
45.6 
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174287 
174014 
173741 

55630  '83098 

5565483082 
55678  ;830ii6 

12 
11 
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51 
52 
53 

9.745871 
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54 

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919085 

14.1 

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172649 

65776  83001 

6 

55 
56 
57 
58 

746624 
746812 
746999 
7471.87 

31.3 
31.3 
31.3 
31.3 

919000 
918915 
918830 
918745 

14.1 
14.1 
14.2 
14.2 

827624 
827897 
828170 
828442 

16.6 
45.5 
46.4 
45.4 

172376 
172103 
171830 
171558 

55799  82985 
5582382969 
55847  '82953 
5587182936 

6 
4 
3 

2 

69 

747374 

31.2 

918659 

14.2 

828716 

t5.4 

171285 

5589582920 

1 

60 

747562 

J1.2 

918574 

14.2 

828987 

t6.4 

171013 

55919  [82904 

0 

Cosh**. 

Sine. 

Cotang. 

Tang. 

~N7cos.lN.sine. 

~ 

56  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (34°)  Natural  Sines.        *~"*33~~ 

' 

Sine. 

D.  10' 

Cosine. 

D.  10' 

Tang. 

D.  10 

Cotang. 

N  .sine 

N.  cos.| 

0 

9.747562 

9.918574 

9.828987 

10.171013 

56919 

82904 

60 

1 

747749 

31  .2 

918489 

14.2 

829260 

45.4 

170740 

55943 

82887 

59 

2 

747936 

31  .2 

918404 

14.  i 

829532 

46  .4 

170468 

55968 

82871 

68 

3 

748123 

31  .2 

918318 

14.5 

829805 

46.4 

170195 

65992 

82856 

57 

4 

748310 

31  .1 

918233 

14.5 

830077 

45.4 

169923 

56016 

8283S 

56 

5 

748497 

31.1 

918147 

14. 

1  1 

830349 

45.4 

169651 

56040 

82822 

56 

6 

748683 

31  .1 

918062 

14. 

i  i 

830621 

45.  c 

169379 

56064 

82806 

54 

7 

748870 

31.1 

917976 

14.  i 

830893 

45.  c 

169107 

56088 

82790 

53 

8 

749056 

31.1 

917891 

14. 

831165 

45.  c 

168835 

56112 

82773 

52 

9 

749243 

31.0 

917805 

14. 

831437 

45.  c 

168563 

66136 

82757 

61 

10 

749426 

31.0 

Ol  A 

917719 

14.  < 

1  A  ' 

831709 

46.  c 

4n  ^ 

168291 

56160 

82741 

60 

11 

9.749615 

dl  .U 

o  t  n 

9.917634 

14.  i 

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9.831981 

AK.  c 

10.168019 

66184 

82724 

49 

12 

749801 

dl  .  U 

917548 

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832263 

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167747 

56208 

82708 

48 

13 

749987 

31  .0 
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917462 

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Ho 

832525 

45.  c 

167476 

66232 

82692 

47 

14 

750172 

du.y 

917376 

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1  A  o 

832796 

45  .  c 

167204 

56256 

82675 

46 

15 

750358 

30.9 

917290 

14.  d 

1  A  O 

833068 

45.  c 

166932 

56280 

82659 

46 

16 

750543 

30.9 

917204 

14.  d 

1/1   O 

833339 

46  .'. 

166661 

66305 

82643 

44 

17 

750729 

30.9 

917118 

14.  d 

MA 

833611 

45  . 

166389 

56329 

82626 

43 

18 

750914 

30.9 

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833882 

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752760  ^'L 

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t  A  K. 

836593 

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163407 

56593 

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762944  XX'  « 

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915994 

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162325 

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837946 

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162054 

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82363 

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838216 

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161784 

56736 

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44.  o 

A  A  Q 

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1 

60 

758591  ;  d 

913365 

14.7 

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44.  o 

154773 

57358 

81915 

0 

Cosine. 

Sine. 

Cotang. 

Tang.  .) 

N.  cos. 

N.siue. 

' 

65  Degrees. 

56          Log.  Sines  and  Tangents.  (35°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

1>.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotaug. 

N.  sine. 

N.  cos. 

0 

9.758591 

Of)  1 

9.913365 

14.  7 

9.845227 

4.4.  S 

10.154773 

57358 

81915 

60 

j 

758772 

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913276 

14  .  i 

845496 

44.  O 

A  A   Q 

154504 

57381 

81899 

59 

2 
3 

758952 
769132 

30.  0 
30.0 

913187 
913099 

14.7 
14.8 

845764 
846033 

44.0 

44.8 

164236 
.   153967 

57405 
67429 

81882 
81866 

58 
67 

4 

759312 

30.0 

913010 

14.8 

•t  4   O 

846302 

44.8 

153698 

57463 

81848 

56 

6 

759492 

30.0 

912922 

14.8 

UCl 

846570 

44.8 

153430 

67477 

81832 

55 

6 

759672 

30.0 

on  (i 

912833 

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1  A  Q 

846839 

44.7 

153161 

57601 

81815 

54 

7 

759852 

ii9.9 

912744 

14.0 

•t  A  Q 

847107 

44.7 

t  A   17 

152893 

57524 

81798 

53 

8 

760031 

29.  9 

Cyr\  Q 

912655 

14.0 

11   Q 

847376 

44.  / 

A  A  n 

152624 

67548 

81782 

52 

9 

760211 

2y  .  y 

912666 

14.0 

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847644 

44.  / 

152356 

67572 

81765 

51 

10 

760390 

29.9 

912477 

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t  4  Q 

847913 

44.7 

152087 

67596 

81748 

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9.760569 

29.9 

9.912388 

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9.848181 

44.7 

10.151819 

57619 

81731 

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12 

760748 

29.8 

912299 

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44.7 

151551 

67643 

81714 

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13 

760927 

29.  8 

912210 

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44.7 

151283 

57667 

81698 

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761106 

29.  8 

912121 

14.9 

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44.7 

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67691 

81681 

46 

15 

761285 

29.8 

912031 

14.9 

849254 

44.7 

150746 

6771581664 

45 

16 
17 

761464 
761642 

29.8 
29.8 

911942 
911853 

14.9 
14.9 

849522 
849790 

44.7 
44.7 

150478 
160210 

5773881647 
57762|81631 

44 

43 

18 

761821 

29.7 

911763 

14.9 

860058 

44.6 

149942 

157786181614 

42 

19 

761999 

29.7 

911674 

14.9 

860325 

44.6 

149675 

5781081697 

41 

20 

762177 

29.7 

911584 

14.9 

1  A  Cl 

850593 

44.6 

A  A   (Z 

149407 

15783381580 

40 

21 

22 

9.762356 
762534 

29.7 
29.7 

9.911495 
911405 

14.9 
14,9 

9.850861 
851129 

44.0 
44.6 

10.149139 

148871 

5785781563 
5788181546 

39 

38 

23 

762712 

29.6 

911315 

14.9 

851396 

44.6 

148604 

5790481530 

37 

24 

762889 

29.6 

911226 

15.0 

851664 

44.6 

148336 

6792881513 

36 

25 

763067 

29.6 

911136 

15.0 

-I  ~  A 

861931 

44.6 

148069 

6795281496 

35 

26 

763245 

29.6 

911046 

lo.O 

852199 

44.6 

147801 

6797681479 

34 

27 

763422 

29.6 

910956 

15.0 

852466 

44.6 

147534 

5799981462 

33 

28 

763600 

29.  6 

910866 

15.0 

1  K  A 

852733 

44.6 

147267 

5802381445 

32 

29 

763777 

29.5 

910776 

16.0 

853001 

44.5 

146999 

5804781428 

31 

30 

763954 

29.  5 

910386 

15  .0 

853268 

44.5 

146732 

5807081412 

30 

31 

9.764131 

29.5 

9.910596 

15.  0 

9.853535 

44.6 

10-146465 

6809481395 

29 

32 

764308 

29.5 
oo  & 

910506 

15.0 
1  Pi  ft 

853802 

44.5 

A  A  K 

146198 

5811881378 

28 

33 

764485 

AiJ  .  0 

910415 

10  .  u 

854069 

44  .  O 

145931 

58141  81361 

27 

34 

764662 

29.4 

910325 

15.0 

854336 

44.5 

145664 

5816581344 

26 

35 

764838 

29.4 

910236 

16.1 

854603 

44.5 

145397 

5818981327 

25 

36 
37 

765015 
765191 

29.4 
29.4 

910144 
910054 

15. 
15. 

864870 
855137 

44.5 
44.5 

145130 

144863 

58212J81310 
58236  81293 

24 
23 

38 
39 

765367 
765544 

29.4 
29.4 

909963 
909873 

15. 
15. 

865404 
855671 

44.5 
44.5 

144596 
144329 

5826081276 
58283181259 

22 
21 

40 

765720 

29.  3 

909782 

15. 

855938 

44.4 

144062 

58307. 

81242 

20 

41 

9.765896 

29.3 

9.909691 

15. 

9.856204 

44.4 

10-143796 

58330 

81225 

19 

42 

766072 

29.3 

909601 

15. 

856471 

44.4 

143529 

58354 

81208 

18 

43 

766247 

29.3 

909510 

16. 

856737 

44.4 

143263 

58378 

81191 

17 

44 

766423 

29.3 

909419 

15. 

857004 

44.4 

142996 

58401 

81174 

16 

45 

766598 

29.3 

909328 

L5. 

857270 

44.4 

142730 

58425 

81157 

15 

46 

766774 

29.2 

909237 

16.2 

i  e  O 

857537 

44.4 

142463 

58449 

81140 

14 

47 

766949 

29.2 

909146 

16.  * 

1  K.  O 

857803 

44.4 

142197 

58472 

81123 

13 

48 

767124 

29.2 

909055 

15.^ 

858069 

44.4 

141931 

58496 

81106 

12 

49 

767300 

29.2 

908964 

15.2 

1  "  O 

858336 

44.4 

141664 

58519 

81089 

11 

50 

767476 

29.2 

OO,  1 

908873 

lo.^ 
1  F>  1 

858602 

44.4 

A  A   0 

141398 

58543 

81072 

10 

61 

9.767649 

^y.  i 

9.908781 

LO  .4 

9.858868 

44  .  o 

10-141132 

58567 

81055 

9 

52 

767824 

29.1 

908690 

15.2 

859134 

44.3 

140866 

58590 

81038 

8 

53 

767999 

29.1 

9085  j9 

16.2 

859400 

44.3 

A  A   Q 

140600 

68614 

81021 

7 

64 

768173 

29.1 

908507 

15.2 

859666 

44.0 

A  A  Q 

140334 

58637 

81004 

6 

65 

768348 

29.1 

908416 

15.2 

859932 

44.0 

140068 

58661 

80987 

5 

66 

768622 

29.0 

908324 

15.3 

860198 

44.3 

139802 

58684 

80970 

4 

57 

768697 

29.0 

908233 

15.3 

860464 

44.3 

139536 

68708 

80953 

3 

58 

768871 

29.0 

908141 

15.3 

860730 

44.3 

139270 

58731 

80y36 

2 

59 

769046 

29.0 

908049 

15.3 

860995 

44.3 

139005 

68755 

80919 

1 

60 

769219 

29.0 

907958 

15.3 

861261 

44.3 

138739 

58779 

8U902 

0 

Cosine. 

Sine. 

Cotang. 

Tang.   ||N.  cos. 

-N  .sine. 

' 

54  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (36°)  Natural  Sines. 

57 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang. 

N.  sine. 

N.  cos. 

0 

9.769219 

9.907958 

9.861261 

A  A  O 

10.138739 

58779 

80902 

60 

1 

769393 

29.0 

907866 

16.3 

861527 

44.  o 

138473 

58802180885 

59 

2 
3 

769566 
769740 

28.9 
28.9 

907774 
907682 

15.3 
15.3 

1C  Q 

861792 
862058 

44.3 

44.2 

138208 
137942 

5882680867 
!  58849|80850 

58 
57 

4 

769913 

28.9 

907590 

lo.o 

862323 

44.2 

137677 

i  58873J80833 

56 

6 

770087 

28.9 

907498 

15.3 

862589 

44.2 

137411 

!  58896  808  16 

55 

6 

770260 

28.9 

907406 

15.3 

862854 

44.2 

137146 

i  58920  80799 

54 

7 

770433 

28.8 

907314 

15.3 

863119 

44.2 

136881 

;  68943  80782 

53 

8 

770606 

28.8 

907222 

15.4 

863385 

44.2 

136615 

58967  80766 

52 

9 
10 

770779 
770952 

28.8 
28.8 

OU  Q 

907129 
907037 

16.4 
15.4 

863650 
863915 

44.2 
44.2 

136360  i  5899080748 
136085  i  69014  80730 

51 
50 

11 

9.771125 

£O  .  O 
OQ  0 

9.906945 

15.4 

1  K  4 

9.864180 

44.2 

10.135820  69037 

80713 

49 

12 

771298 

*>o  .  o 

906852 

10.4 

864445 

44.  Z 

135555  59061 

80696 

48 

13 

771470 

28.7 
28  7 

906760 

15.4 

1K,  4 

864710 

44.2 

AA  O 

136290!  5908480679 

47 

14 

771643 

OQ  rr 

906667 

10  .  4 

864975 

44  .  Z 

A  A  -t 

135025  j  59108  80662 

46 

15 

771815 

•*O,7 

OQ  7 

906575 

1  c  4 

865240 

44.1 
44  1 

134760  i  169131  80644 

45 

16 
17 

771987 
772159 

^O  .  / 

28.7 

906482 
906389 

10  .  ^ 

15.4 

1C   C. 

865605 
865770 

44.  1 

44.1 

134495  1  69164  80627 
134230  1  69178J80610 

44 
43 

18 
19 
20 

772331 
772503 
772675 

23.7 
28.6 
28.6 

OQ  a 

906296 
906204 
906111 

15.5 
15.5 
16.6 

IK   P. 

866035 
866300 
866564 

44.1 
44.1 
44.1 

133965 
133700 
133436 

59201  80593 
!  69225180576 
|  59248  80568 

42 
41 
40 

21 

9.772847 

I^o.b 

9.906018 

lo.o 

1C  K 

9.866829 

44.1 

10.133171 

j  69272  80541 

39 

22 
23 

773018 
773190 

2s!e 

OQ  c 

905926 
905832 

lo.o 
15.5 
ic,  5 

867094 
867368 

44.1 
44.1 

A  A  1 

132906 
132642 

1  59295  80524 
69318J80507 

38 
37 

24 

773361 

Z<3  ,  D 
OQ  C 

905739 

10  .  o 

867623 

132377 

69342 

80489 

36 

25 

773633 

zr>  .5 

905645 

15.5 

1  £  C 

867887 

44.1 

132113 

1  59366 

80472 

35 

26 

773704 

28.5 

OQ  C. 

905552 

15.5 
1  p-  c 

868152 

44.1 

131848 

|  59389 

80455 

34 

27 

773875 

-4O.5 

905459 

15.  o 

1  "  C. 

868416 

44.0 

131684 

59412 

80438 

33 

28 

774046 

28.5 

OQ  K 

905366 

1O.5 

868680 

44.0 

131320 

59436|80422 

32 

29 

774217 

Zo  .0 

28  5 

906272 

16.6 
15  6 

868945 

44.0 

AA  A 

131055  i  59459180403 

31 

30 

774388 

OQ   A 

905179 

869209 

44.  U 

130791 

69482 

80386 

30 

31 

9.774558 

iJo.4 

9.905086 

16.6 

9.869473 

44.0 

10.130527 

59506  180368 

29 

32 

774729 

28  4 

904992 

15.6 

1  C  ft 

869737 

44.0 

A  \  A 

130263 

69529  80351 

28 

33 

774899 

904898 

10  .0 

870001 

44.  0 

129999 

59552  80334 

27 

j  34 

775070 

OQ   « 

904804 

15  .6 

1C  ft 

870265 

44.  0 

129735 

i  69576 

80316 

26 

35 

775240 

.*o  .  4 
28  4 

904711 

15  .  o 
i  e  « 

870629 

44.0 

A  A  A 

129471 

69599 

80299 

25 

36 

775410 

28  3 

904617 

IO  .  O 
1C  ft 

870793 

44.  U 

A  A  A 

129207 

69622 

80282 

24 

37 

775580 

28*3 

904523 

IO  .  O 

1  fi  6 

871057 

44.  U 
A  A  A 

128943 

59646 

80264 

23 

38 

775750 

OQ  O 

904429 

10  ,o 

871321 

44  .  U 

128679 

59669 

80247 

22 

39 

775920 

Zo  .3 
28  3 

904335 

15.7 
15  7 

871585 

44.0 

128415  59693 

80230 

21 

40 

776090 

28*3 

904241 

15*7 

871849 

4-3  Q 

128151  169716 

80212 

20 

41 

9.776259 

28*3 

9.904147 

JO.* 

15  7 

9.872112 

10.127888 

69739 

80195 

19 

42 

776429 

904053 

IO  .  1 

1C  1 

872376 

4o  .  y 

127624 

69763 

80178 

18 

43 

776598 

28  9 

903959 

15.  7 
IK  7 

872640 

43.9 

127360 

69786J80160 

17 

44 

776768 

-  o  .  Z 
28  9 

903864 

15  .  / 
1  K  7 

872903 

4n  |\ 

127097 

!  69809J80143 

16 

45 

776937 

-*o  .  -* 

903770 

15  .  i 

873167 

4o.y 

126833 

69832J80125 

15 

46 

777106 

28  .  2 

903676 

15.7 

1C  T 

873430 

43.9 

126570 

59866180108 

14 

47 

777275 

28*1 

903681 

15.  i 
IK  7 

873694 

43.9 

126306]  5987980091 

13 

48 
49 

777444 
777613 

28!l 
28  1 

903487 
903392 

IO  .  / 

15.7 

i  e  Q 

873957 
874220 

43^9 

126043115990280073 
125780||6992680056 

12 
11 

50 

777781 

-o  .  1 

903298 

10  .  o 

1  p-  o 

874484 

43.9 

1256161  169949  180038 

10 

51 

9.777950 

28  1 

9.903202 

lO  .0 

1  K  U 

9.874747 

43.9 

10.  125253  l'6997«|80021 

9 

52 
53 

778119 

778287 

28  '.1 
28  0 

903108 
903014 

IO  ,  O 

16.8 

ICQ 

875010 
875273 

43.  9 
43.9 

/iO  Q 

124990  '  59995  80003 
124727  1  60019|79986 

8 

7 

54 

778455 

28  0 

902919 

IO  .  O 
1C  Q 

876536 

4o  .0 

/iO  W 

124464  j|60042]79968 

6 

55 

778624 

•*o  .  U 

28  0 

902824 

IO  .  O 
1C  Q 

876800 

4o  .  o 

AO  Q 

124200  1  60065 

79961 

6 

56 

778792 

28  0 

902729 

IO  .O 
1C  Q 

876063 

4o  .0 

^0  Q 

123937  60089 

79934 

4 

57 

778960 

^o  .  \3 
28  0 

902634 

IO  .  O 

15  8 

876326 

1  •!   Q 

123674  60112 

79916 

3 

58 

779128 

OQ  A 

902539 

JO  .  O 

15  9 

876589 

4o  .  o 

A.Q  ft 

123411 

160135 

79899 

2 

69 

779295 

-faO  .  U 

27  9 

902444 

15  9 

876851 

4o  .  o 
4Q  Q 

123149 

!  60158 

79881 

1 

60 

779463 

902349 

877114 

4o  .  o 

122886 

|60182 

79864 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

;  N.  cos. 

N'.sine. 

' 

53  Degrees. 

58          Log.  Sines  and  Tangents.  (37°)  Natural  Sines.     TABLE  II. 

1 

Sine. 

D.  10" 

Cosine.  |D.  10" 

Tang. 

D.  10" 

Co  tang. 

N.sine 

N.  cos 

0 

9.779463 

07  q 

9.902349 

1  E  q 

9.877114 

40  o 

10.122886 

60182 

79864 

60 

1 

779631 

*f  i  .  y 

902253 

10  .  y 

877377 

*to  .  o 

122623 

60205 

79846 

59 

2 

779798 

27.9 

902158 

15.9 

1  K  Q 

877640 

43.8 

122360 

60228 

79829 

58 

3 

779966 

27.9 

902063 

lo.y 

877903 

43.8 

122097 

60251 

79811 

57 

4 

780133 

27.9 

901967 

15.9 

878165 

43.8 

A  O  Q 

121835 

60274 

79793 

56 

6 

780300 

27.9 

901872 

15.9 

878428 

4o.o 

/1O  Q 

121572 

60298 

79776 

55 

6 

780467 

27.8 

901776 

15.9 

i  *"  n 

878691 

4o.o 

121309 

60321 

79758 

54 

7 

780634 

27.8 

901681 

lo.y 

1  Ft  Q 

878953 

43.8 

40  7 

121047 

60344 

79741 

53 

8 

780801 

07*8 

901585 

lo.y 

1  n  Q 

879216 

4o  .  / 
,40  7 

120784||60367 

79723 

62 

9 

780968 

£  1  ,  O 

901490 

10  .  y 

879478 

4o  .  i 

120522 

60390 

79706 

51 

10 

781134 

27.8 

901394 

15.9 

879741 

43.7 

120259 

60414 

79688 

50 

11 

9.781301 

27.8 

9.901298 

16.0 

9.880003 

43.7 

An  i*f 

10.119997 

60437 

79671 

49 

12 

781468 

27.7 

901202 

16.  0 

880265 

4o.  / 

119735 

60460 

79658 

48 

13 

781634 

27.7 

901106 

16.0 

880528 

43.7 

119472 

60483 

79635 

47 

14 

781800 

27.7 

901010 

16.0 

880790 

43.7 

An  rj 

119210 

60506  179618 

46 

16 

781966 

27.7 

900914 

16.0 

881052 

4o.  / 

118948 

60529 

79600 

46 

16 

782132 

27.7 

900818 

16.0 

881314 

43.7 

118686 

60553 

79583 

44 

17 

782298 

27.7 

900722 

16.0 

881576 

43.7 

118424 

60576 

79565 

43 

18 

782464 

27.6 

900626 

16.0 

881839 

43.7 

118161 

60599 

79647 

42 

19 

782630 

27.6 

900529 

16.0 

1  G  *  A 

882101 

43.7 

117899 

60622 

79530 

41 

20 

782796 

27.6 

900433 

lo.  U 

882363 

43.7 

117637 

60645 

79512 

40 

21 

9.782961 

27.6 

9.900337 

16.1 

9.882625 

43.6 

10.117375 

60668 

79494 

39 

22 

783127 

27.6 

900242 

16.1 

882887 

43.6 

117113 

60691 

79477 

38 

23 

783282 

27.6 

900144 

16.1 

883148 

43.6 

116852 

60714 

79459 

37 

24 

783458 

27.6 

900047 

16.1 

883410 

43.6 

116590 

60738 

79441 

36 

25 

783623 

27.5 

899951 

16.1 

883672 

43.6 

116328 

60761 

79424 

35 

26 

783788 

27.6 

899854 

16.1 

883934 

43.6 

116066 

60784 

79406 

34 

27 

783953 

27.6 

899757 

16.1 

884196 

43.6 

115804 

60807 

79388 

33 

28 

784118 

27.5 

899660 

16.1 

884457 

43.6 

115543 

60830 

79371 

32 

20 

784282 

27.5 

899564 

16.1 

884719 

43.6 

115281 

60853 

79353 

31 

39 
31 
32 

784447 
9.784612 

784776 

27.4 
27.4 
27.4 

899467 
9.899370 
899273 

16  1 
16'2 
16.  '2 

884980 
9.886242 
885503 

43.6 
43.6 
43.6 

116020  60876 
10.114758!  60899 
114497]  160922 

79335 
79318 
79300 

30 

29 

28 

33 

784941 

27.4 

899176 

16.2 

885766 

43.6 

114236 

60945 

79282 

27 

34 

785105 

27.4 

899078 

16.2 

886026 

43.6 

113974 

60968 

79264 

26 

35 

786269 

27.4 

898981 

16.2 

886288 

43.6 

113712 

60991 

79247 

25 

36 

785433 

27.3 

898884 

16.2 

886549 

43.6 

113451 

61015 

79229 

24 

37 

785597 

27.3 

898787 

16.2 

886810 

43.5 

113190 

61038 

79211 

23 

38 

785761 

27.3 

898689 

16.2 

887072 

43.5 

112928 

61061 

79193 

22 

39 

785925 

27.3 

898692 

16.2 

887333 

43.6 

112667 

61084 

79176 

21 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
61 
62 
63 
64 
65 
66 
67 
58 
59 
60 

786089 
9.786252 
786416 
786679 
786742 
786906 
787069 
787232 
787396 
787567 
787720 
9.787883 
788046 
788208 
788370 
788532 
788694 
788856 
789018 
789180 
789342 

27.3 
27.3 
27.2 
27.2 
27.2 
27.2 
27.2 
27.2 
27.1 
27.1 
27.1 
27.1 
27.1 
27.1 
27.1 
27.0 
27.0 
27.0 
27.0 
27.0 
27.0 

898494 
9.898397 
898299 
898202 
898104 
898006 
897908 
897810 
897712 
897614 
897516 
9.897418 
897320 
897222 
897123 
897025 
896926 
896828 
896729 
896631 
896532 

16.2 
16.3 
16.3 
16.3 
16.3 
16.3 
16.3 
16.3 
16.3 
16.3 
16.3 
16.3 
16.4 
16.4 
16.4 
16.4 
16.4 
16.4 
16.4 
16.4 
16.4 

887694 
9.887855 
888116 
888377 
888639 
888900 
889160 
889421 
889682 
889943 
890204 
9.890466 
890725 
890986 
891247 
891507 
891768 
892028 
892289 
892549 
892810 

43.5 
43.6 
43.5 
43.6 
43.5 
43.5 
43.5 
43.5 
43.5 
43.5 
43.5 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 

112406 
10.112145 
111884! 
111623 
111361 
111100 
110840! 
110679 
110318! 
110057  ! 
109796  ! 
10.109536! 
109275 
109014! 
108753 
108493 
108232 
107972 
107711  i 
107451  i 
107190 

61107 
61130 
61163 
61176 
61199 
61222 
61246 
61268 
61291 
61314 
61337 
61360 
61383 
61406 
61429 
61451 
61474 
61497 
61520 
61543 
61566 
—  

79168 
79140 
79122 
79105 
79087 
79069 
79051 
79033 
79016 
78998 
78980 
78962 
78944 
78926 
78908 
78891 
78873 
78855 
78837 
78819 
78801 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

Cosine. 

Sine. 

Co  tang. 

Tang.   i  IN.  cos. 

N.sine. 

' 

52  Degrees. 

TABLE  H.     Log.  Sines  and  Tangents.  (38°)  Natural  Sines. 

/      c;-rhA     T»  1<V/   rVio»T»/i    Tl  1A"    Tanrr    "H  1  fVI   (\\tartcr    f  "N  aino 

59 

0 

.789342 

.896532 

^w*e* 

9.892810 

10.107190! 

61566 

78801 

60 

1 

789504 

26.9 

896433 

16.4 

893070 

43.4 

106930  '• 

61589 

78783 

59 

2 

789665 

26.9 

896335 

L6.5 

893331 

43.4 

106669 

61612 

78765 

68 

3 
4 
5 

789827 
789988 
790149 

26.9 
26.9 
26.9 

896236 
896137 
896038 

16.6 
16.6 
16.5 

1  a  K 

893591 
893851 
894111 

43.4 
43.4 
43.4 

,40    A 

106409 
106149! 
105889 

61636 
61658 
61681 

78747 
78729 
78711 

67 
56 
55 

6 

790310 

26.9 

895939 

lo.o 

1  />   K. 

894371 

4o.4 

jo  x 

105629  1 

61704 

78694 

54 

7 

790471 

26.8 

895840 

lo.o 

894632 

4o.4 

105368  | 

61726 

78676 

53 

8 
9 

790632 
790793 

26.8 
26.8 

895741 
896641 

16.5 
16.6 

1  C  K. 

894892 
895152 

43.3 
43.3 

105108 

104848 

61749 
61772 

78668 
"8640 

52 
51 

10 

790954 

26.8 

895642 

16.  o 

1  £2  K. 

895412 

43.3 

104688 

61795 

78622 

50 

11 
12 

9.791115 

791275 

26.8 
26.8 

9.895443 
895343 

lo.o 
16.6 
ic  A 

9.895672 
895932 

43  !s 

A  0   O 

10.1043281 
104068 

61818 
61841 

78604 
78586 

49 

48 

13 
14 

791436 
791596 

26^7 

895244 
895145 

lo.o 
16.6 

1  d   fi 

896192 
896452 

4o  .0 
43.3 

103808 
103548  i 

61864 
61887 

78568 
78550 

47 
46 

15 

791757 

26.7 

895045 

10  .  o 

896712 

43.3 

103288 

61909 

78632 

45 

16 

791917 

26.7 

894946 

16.6 

896971 

43  .3 

103029  ! 

61932 

78514 

44 

17 

792077 

26.7 

894846 

16.6 

1  £»  £ 

897231 

43.3 

102769 

61955 

78496 

43 

18 

792237 

26.7 

894746 

lo.o 
If!  8 

897491 

43.3 

40  o 

102509  j  61978 

78478 

42 

19 

792397 

26.6 

894646 

10  .  o 

10.  A 

897751 

4i>  .  o 

102249  162001 

78460 

41 

20 

792557 

26  .  6 

894546 

lo.o 

1  0.  p. 

898010 

43.3 

101990  162024 

"8442 

40 

21 

9.792716 

26.6 

9.894446 

lo.o 

9.898270 

43-3 

10.101730 

62046 

78424 

39 

22 

792876 

26.6 

894346 

16.7 

10  T 

898530 

43.3 

101470 

62069 

"8405 

38 

23 

793035  I*-'? 

894246 

16.7 

10  rr 

898789 

43-3 

101211 

62092 

"8387 

37 

24 

793195  ^'r 

894146 

16.7 

899049 

43-3 

100951 

62115 

"8369 

36 

25 

793354 

894046 

16.7 

899308 

43  .2 

100692 

62138 

78351 

35  . 

26 

793514 

26  .5 

893946 

16.7 

899568 

43  -2 

100432 

62160 

78333 

34 

27 

793673i£-? 

893846 

16.7 

899827 

43.2 

100173 

62183 

78315 

33 

28 

793832 

zo.o 

893745 

16.7 

900086 

43.2 

099914 

62206 

78297 

32 

29 

793991 

26.5 

rSo   ^ 

893645 

16.7 

900346 

43  .2 

099664 

62229 

78279 

31 

30 

794150  if™ 

893644 

16.7 

900606 

43  .2 

099395 

62251 

78261 

30 

31 

9.  794308  Jo°', 

9.893444 

16.7 

10  Q 

9.900864 

43  .2 

10.099136 

62274 

78243 

29 

32 

794467  i  oft" 

893343 

It)  .  o 

10  U 

901124 

40  o 

098876 

62297 

78225 

28 

33 

7946261  op  'T 

893243 

lO  .  O 
1o  o 

901383 

4O  .  .4 
An  c\ 

098617 

62320 

-8206 

27 

34 

794784  j^'' 

893142 

lo.o 

901642 

4o  .2 

098368 

62342 

78188 

26 

35 
36 

794942  fr' 
795101  ;  or 

893041 
892940 

16.8 
16.8 

901901 
902160 

43.2 
43.2 

098099  1 
097840 

62365 
62388 

78170 

78162 

25 

24 

37 

795259  OR  o 

892839 

16.8 

1C*  U 

902419 

43.2 

097681 

62411 

78134 

23 

38 

795417  op"  o 

892739 

16.  0 
IK  K 

902679 

43.2 

AQ  O 

097321 

62433 

78116 

22 

39 

795575  OK'O 

892638 

lO.  o 
1ft  R 

902938 

^±0  .^ 

AQ  O 

097062 

62456 

78098 

21 

40 

795733  oTo 

892536 

10  .  o 

903197 

"*«J  •  -^ 

096803 

62479 

78079 

20 

41 

9.795891  O«'Q 

9.892435 

16.8 

9.903455 

43.1 

10.096545 

62602 

78061 

19 

42 

796049  oTo 

892334 

16.9 
i  o.  n 

903714 

43.1 

096286 

62524 

78043 

18 

43 

796206  o°'o 

892233 

io.y 

903973 

43.1 

096027 

62547 

78025 

17 

44 

796364  OK"; 

892132 

16.9 

904232 

43.1 

095768 

62670 

78007 

16 

45 

796521  JJ-J 

892030 

16.9 

10  O 

904491 

43.1 
40  i 

095509 

62592 

77988 

15 

46 

796679  o"'r 

891929 

10  ,y 

10   Q 

904750 

io  .  l 

095260 

62616 

77970 

14 

47 

796836  o^;: 

891827 

10  .  y 

905008 

"  •  1 

094992 

62638 

77962 

13 

48 

796993  oro 

891726 

16.9 

905267 

43.1 

094733 

62660 

77934 

12 

49 

797160  o«  i 

891624 

16.9 

10   Q 

905626 

43.1 

094474 

62683 

77916 

11 

50 

797307  o« 

891623 

1O  .b 

906784 

AQ 

094216 

62706 

77897 

10 

51 

9.797464  £': 

9.891421 

17.  (J 

nf] 

9.906043 

40  i 

10.093967 

62728 

77879 

9 

52 

797621  ~°' 

891319 

.  u 

906302 

4tO  .  1 

093698 

62761 

77861 

8 

53 

797777  OR 

891217 

17.  (i 

906560 

43.1 

093440 

62774 

77843 

7 

54 

797934  ~' 

891115 

17.  C 

906819 

43.1 

093181 

62796 

77824 

6 

55 

798091  £' 

8910U 

17.0 

nr 

907077 

43.1 
40  i 

092923 

62819 

77806 

6 

66 

798247  *' 

890911 

.U 
nr 

907336 

4u  .  1 

092664 

62842 

77788 

4 

57 

798403  g'J 

890809 

.  t 

1-7  r 

907594 

43  1 

092406 

62864 

77769 

3 

58 

798560  or 

890707 

1  «1 

907852 

092148 

62887 

77751 

2 

69 

798716  g'J 

890605 

17.  C 

908111 

43.1 

091889 

62909 

77733 

1 

60 

798872  26'° 

890503 

17.  t 

908369 

43.0 

091631 

62932 

77716 

0 

Cosine. 

Sine. 

Cotang. 

Tang.   |  |N.  cos 

N.sine 

~ 

ftl  Degrees. 

20 


Log.  Stnos  and  Tangents.  (39°)  Natural  Sines.     TABLE  II. 

Sine. 

D.  l(x 

Cosine. 

D.  10 

Tang. 

D.  10 

Cotang. 

|N.  sin 

N.  C08.| 

9.79877S 

2fi 

9.89050 

9.908369 

1O 

10.091631 

6293 

77715 

60 

799028 

[•6O  . 

26 

89040 

. 

908528 

4o. 

1  O 

091372 

6295 

77696 

59 

799184 

Ofi' 

890298 

. 

908886 

4o. 

40  t 

091114 

6297 

77678 

58 

799339 

25' 

89019 

. 

909144 

•*o  . 
/io 

090856 

6300 

77660 

57 

799495 

»^O  • 
OX   Q 

89009 

. 

909402 

4o  . 

090598 

6302 

77641 

66 

799651 

^O  • 

ox 

•  88999 

. 

909660 

43. 

40   A 

090340 

6304 

77623 

65 

799806 

^O  • 
9n  Q 

88988 

B 
17 

909918 

4o.  ' 

.40  r\ 

090082 

63066 

77605 

64 

799962 

*O  • 
OX  ( 

88978 

li» 

17 

910177 

4o.  i 

AQ  i 

089823 

6309 

77586 

53 

800117 

£d  .  v 
OK   ( 

889682 

!'• 

17 

910435 

4o.  ' 
AQ  n 

089565 

631K 

77568 

52 

800272 

xO  1  5 
ox 

889679 

1  1  • 

910693 

4o  ,  ' 

1Q  i 

089307 

6313E 

77650 

51 

1 

800427 

-*O  * 
ox 

88947 

. 

910951 

*xO  ,  ' 

.40   1 

089049 

6315fc 

77531 

50 

] 

9.800582 

*Q» 

OK 

9.889374 

. 

9.911209 

4o  .  i 

40   A 

10-088791 

9318C 

77513 

49 

12 

800737 

zo  • 

OX 

88927 

•  ** 

nt 

911467 

*lo  .  I 

40    f\ 

088533 

63203 

77494 

48 

13 

800892 

^o.  o 
ox  c 

889168 

,*> 

ns 

911724 

4o  .  1 
40  n 

088276 

63225 

77476 

47 

14 

801047 

^o  .  c 

ox  c 

889064 

.* 
nt 

911982 

4o.  ( 

40   A 

088018 

63248 

17458 

46 

15 

801201 

^D  •  c 

ox  t 

888961 

.A 

179 

912240 

4o.  ( 

AQ  A 

087760 

63271 

77439 

45 

16 

801356 

^O  •  c 

ox  »• 

888858 

I  /  .  / 

nc 

912498 

4o.  1 

40   A 

087502 

63293 

77421 

44 

17 

801511 

-iO  •  , 
ox  • 

888755 

.^ 
nn 

912756 

4o.  ( 
40  n 

087244 

63316 

77402 

43 

18 

801665 

*o  •  , 
ox  r 

888651 

.  * 
ncy 

913014 

4o.  ( 

XO  Q 

086986 

63338 

77384 

42 

19 

801819 

-*O  •  i 
ox  p 

888548 

•  i 

no 

913271 

4^.y 

Af)   Q 

086729 

63361 

77366 

41 

20 

801973 

,*O  •  i 
o"  n 

88H444 

,  ^ 

nM 

913529 

4-^.i 

086471 

6338b 

77347 

40 

21 

9.802128 

I6O.  7 
ox,  7 

9.888341 

.i 

no 

9.913787 

42.  i 

J.O  Q 

10-036213 

63405 

77329 

39 

22 

802282 

^O  •  i 
On  fi 

888237 

.  i 
no 

914044 

H-i  ,  t 

4O  A 

085956 

63428 

77310 

38 

23 

802436 

*0  •  I 

ox  (? 

888134 

.  < 

17  I 

914302 

4-«  .  C 
AO  Q 

085698 

63451 

77292 

37 

24 
26 

802589 
802743 

^O  •  I 

25.6 

O"  A 

888030 
887926 

17.1 

no 

914560 
914817 

T;-«J  .  i: 

42.9 

085440 
086183 

63473 
63496 

77273 

77255 

36 
35 

26 

8028^7 

jiO.O 
OK  K 

887822 

.  i 

170 

915075 

45^.  t 

1*>  O 

084925 

6351b 

77236 

34 

27 

803050 

-4O.  O 

ox  P 

887718 

Lf.ii 

915332 

4^j.y 

10  o 

084668 

63540 

77218 

33 

28 

803204 

^O-O 
OX  A 

887614 

L7.J 

17  '- 

915590 

*±Z  .  c 
49  Q 

084410 

6356b 

77199 

32 

29 

803357 

^o.o 

OK  K. 

887510 

7*r 

915847 

4^s.  y 
J.9  O 

084163 

63585 

77181 

31 

30 

803511 

zo.o 

OX  X 

887406 

i  '  .0 
7  J. 

916104 

•i^.y 

<O  Q 

033896 

63603 

77162 

30 

31 

.803664 

xiSO.  0 

O"  X 

.887302 

-  •  .  4 

7.1 

.916362 

•u.y 

/1O  n 

10-083638 

63630 

7144 

29 

32 

803817 

Vo.o 

O-  K 

887198 

B  4 
7. 

916619 

».S 

033381 

63653 

7125 

28 

33 

803970 

iio.  o 

887093 

.  ^ 

7.4 

916877 

t2.9 
2^-^ 

083123 

63675 

7107 

27 

34 

804123 

25.5 

886989 

.  4 

7,1 

917134 

.9 

1  0  Ci 

082866 

63698 

7088 

26 

35 

804276 

25-  5 

88G885 

.^ 

917391 

t^.9 

082609 

63720 

7070 

2o 

36 

804428 

25.4 

886780 

7.^ 

917648 

^2.9 

082352 

6374-2 

7051 

24 

37 

804581 

25-4 

886676 

T  .4 

917905 

i2.9 

082095 

637b5 

7033 

23 

38 

804734 

25.4 

886571 

T  .4 

918163 

=2.9 

081837 

63787 

7U14 

22 

39 

804886 

25-4 

886456 

7.4 

918420 

:2.8 

081580 

63810 

6996 

21 

40 

805039 

25-4 

OX  A 

886362 

7.4 
7  f\ 

918677 

42.8 

O  Q 

08  13231  (63832 

6977 

29 

41 

.805191 

20.** 

•886257 

i  .  O 

7[T 

.918934 

4^.O 
O  Q 

0-081066  63854 

6959 

19 

42 

805343 

25.4 

OK  Q 

886152 

.6 

7er 

919191 

Se.o 

0808091  63877 

6940 

18 

43 
44 

805495 
805647 

25  -o 
25.3 

OK  Q 

886047 
885942 

.O 

7.5 

7cr 

919448 
919705 

2.8 

42.8 

/1O  Q 

080552  | 
030295 

63899 
63922 

6921 
J903 

17 
16 

45 

805799 

25.  o 

885837 

.O 

919962 

4-^.0 

080038 

63944 

J884 

15 

46 

805951 

25.3 

885732 

7.  5 

920219 

42.8 

079781  1 

6396b 

b866 

14 

47 

806103 

25.3 

885627 

7.5 

920476 

42.8 

079524 

6398;; 

6847 

13 

48 

806254 

25.3 

OX  Q 

885522 

7.5 

7X. 

920733 

42.8 

AC)  Q 

079267  1 

64011 

6828 

12 

49 

806406 

zo.o 

OK  9 

885416 

.  o 

7C 

920990 

*±6.  O 

AC)   Q 

079010  164033 

'6810 

11 

50 

806557 

^O.^ 

885311 

•  o 

921247 

4x6.  O 

078753  164056 

76791 

10 

51 

.806709 

25.2 
ox  o 

.885205 

7.6 

7  fi 

.921503 

42.8 
10  ft 

0-078497  j 

64078 

r6772 

52 

806860 

zo  .* 
ox  o 

885100 

/  .  o 

7fJ 

921760 

4Z.  O 
AO  W 

078240  | 

64100 

r6764 

8 

53 
54 

807011 
807163 

J'-)  •  ~ 

25.2 

834994 
884889 

.  o 

7.6 

922017 
922274 

4^  .  o 

42.8 

077983  ! 
077726  | 

64123 
64145 

?6735 
r6717 

I 

55 

807314 

25.2 

884783 

7.6 

922530 

42.8 

/iO  U 

077470  .' 

64167 

'6698 

5 

56 
57 

807465 
807615 

25.2 
25.1 

884677 
884572 

7.6 
7.6 

922787 
923044 

4!^.o 
42.8 

077213,  (64190 
076956!  642  11: 

^6679 
'6661 

4 

58 

807766 

25.1 

884466 

7.6 

923300 

42.8 

076700  i 

64234 

6642 

2 

59 

807917 

25.1 

884360 

7.6 

923557 

42.8 

076443 

64256 

6623 

I 

60 

808067 

25.  1 

884254 

7.6 

923813 

42.7 

076187  | 

64279 

6604 

o 

Cosine. 

Sine. 

Cotang. 

Tan,g.   ;  |N.  coa. 

x'.piiie. 

50  Degrees. 

TABLE  H.     Log.  Sines  and  Tangents.  (40°)  Natural  Sines.          61 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang. 

N  .sine. 

N.  cos. 

0 

9.808067 

9.884254 

nry 

9.923813 

10.076187 

64279 

76604 

60 

1 

808218 

OK  1 

884148 

.  / 

nrv 

924070 

497 

076930 

64301 

76586 

59 

2 

808368 

,*O  .  I 
9fi  1 

884042 

.  1 
nri 

924327 

49  7 

075673 

64323 

76567 

68 

3 

808519 

>*0  .  1 
9K  ft 

883936 

.  I 
irr  ri 

924683 

49  7 

076417 

64346 

76548 

57 

4 

808669 

•iO  .  \J 
OK  A 

883829 

1  '  .  ' 

924840 

4-4.7 

075160 

64368 

76630 

56 

6 

808819 

2o.U 

883723 

17.7 

925096 

A 

074904 

64390 

76511 

55 

6 

808969 

OK  A 

883617 

17.7 

925352 

49  'ft 

074648 

64412 

76492 

54 

7 

809119 

-4O.  U 
OK  A 

883510 

17.7 

925609 

A 

074391 

64435 

76473 

53 

8 

809269 

;4o.u 

OK  A 

883404 

17.7 

i  ~  n 

925865 

42.7 

074135 

64457 

76455 

62 

9 

809419 

4O  .  U 
O  4   Q 

883297 

17.7 

no 

926122 

42.7 

073878 

64479 

76436 

51 

10 

809569 

^4.  y 

O/4  Q 

883191 

.0 

•\  1  O 

926378 

49  7 

073622 

64601 

76417 

50 

11 

9.809718 

24.  y 

9.883084 

17.0 

no 

9.926634 

AC)  >7 

10.073366 

64524 

76398 

49 

12 

809868 

24.  9 

94.  Q 

882977 

.0 

1  1  Q 

926890 

4,6  «  1 

Act  n 

073110 

64546 

76380 

48 

13 

810017 

24.  y 

882871 

17.0 

1  T  Q 

927147 

<±4  ,  I 

072853 

64568 

76361 

47 

14 

810167 

94.  *Q 

882764 

17.0 

•t  1  Q 

927403 

49  7 

072597 

64690 

76342 

46 

15 

810316 

;*4.y 

882667 

17.0 

n  nr  Q 

927659 

49  7 

072341 

64612 

76323 

46 

16 

810465 

•>A  8 

882550 

17.0 

927915 

497 

072085 

64635 

76304 

44 

17 

810614 

8 

882443 

17.8 

928171 

A 

071829 

64657 

76286 

43 

18 

810763 

o 

882336 

17.8 

928427 

42.7 

071573 

64679 

76267 

42 

19 

810912 

24.8 

882229 

17.9 

928683 

42.  7 

071317 

64701 

76248 

41 

20 

811061 

f)A  °8 

882121 

17.9 

928940 

42.7 

071060 

64723 

76229 

40 

21 

9.811210 

24.  o 
o/i  8 

9.882014 

17.9 

9.929196 

42.7 

10.070804 

64746 

76210 

39 

22 

811358 

24.o 
94  7 

881907 

17.9 

929452 

42.7 

070548 

64768 

76192 

38 

23 

811507 

24.  / 

9/4  7 

881799 

17.9 

929708 

97 

070292 

64790 

76173 

37 

24 

811655 

24.  i 

9/1  7 

881692 

17.9 

929964 

AC\'  a 

070036 

64812 

76154 

36 

25 

811804 

24.  • 

9  \  7 

881584 

17.9 
nr\ 

930220 

42.  0 

069780 

64834 

76135 

35 

26 

811952 

.44.  ' 

881477 

.y 

1  n  o 

930476 

42.6 

069625 

64856 

76116 

34 

27 

812100 

24.7 

881369 

17  .y 

930731 

42  .  6 

069269 

64878 

76097 

33 

28 

812248 

24.7 

881261 

17.9 

930987 

o 

069013 

64901 

76078 

32 

29 

812396 

24.7 

881153 

18.0 

931243 

42.6 

068757 

64923 

76059 

31 

30 

812544 

24.6 

881046 

18.0 

931499 

49  r 

068561 

64945 

76041 

30 

31 
32 

9.812692 
812840 

24.6 
24.6 

94  fi 

9.880938 
880830 

18.0 
18.0 

9.931755 
932010 

42!e 

10.068245 
067990 

64967 
64989 

76022 
76003 

29 

28 

33 

812988 

24.  D 

O/I  ft 

880722 

18  .  0 

1  Q  A 

932266 

49  "ft 

067734 

65011 

75984 

27 

34 

813135 

24.  D 

880613 

lo.U 

932522 

42.6 

067478 

65033 

75965 

26 

36 

813283 

24.6 

880505 

18.0 

932778 

42.6 

067222 

65056 

75946 

25 

36 

813430 

24.6 

880397 

18.0 

933033 

42.6 

066967 

65077 

75927 

24 

37 

813578 

24.5 

880289 

18.0 

933289 

42.  o 

066711 

65100 

75908 

23 

38 
39 

813726 
813872 

24.5 
24.5 

O  4   £ 

880180 
880072 

18. 
18. 

933545 
933800 

42!e 

066455 
066200 

65122 
65144 

75889 
76870 

22 
21 

40 
41 

814019 
9.814166 

24.5 
24.5 

0,4  e 

879963 
9.879856 

18. 
18. 

934056 
9.934311 

42^6 

065944 
10.065689 

65166 
65188 

75861 

75832 

20 
19 

42 

814313 

24.6 

O  \   " 

879746 

18. 

934567 

42.6 

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065433 

65210 

75813 

18 

43 

814460 

24  .  :j 

879637 

18. 

934823 

42  .  o 

ACt  ft 

065177 

65232 

75794 

17 

44 

814607 

O  4   A 

879529 

18. 

935078 

42.  0 

>IO  ft 

064922 

65254 

76775 

16 

45 

814753 

-44.4 

879420 

18. 

935333 

42.  o 

/1O  ft 

064667 

65276 

75(56 

15 

46 
47 

48 

814900 
815046 
815193 

24.4 
24.4 
24.4 

879311 
879202 
879093 

18. 
18. 
18.2 

935589 
935844 
936100 

42.  D 

42.6 
42.6 

ACt   ft 

064411 
064166 
063900 

65298 
65320 
65342 

76738 
75719 
75700 

14 
13 

12 

49 

815339 

24.4 

O1   A 

878984 

18.2 

936355 

42.  o 

063645 

65364 

75680 

11 

50 

815485 

24.4 

O  A   Q 

878875 

18.2 

936610 

42.6 

063390  |i  66080 

76661 

10 

61 

9.815631 

24.  o 

9.878766 

18.2 

9.936866 

42.6 

10.063134 

6540S 

75642 

y 

62 

815778 

24.3 

878656 

18.2 

937121 

42.5 

062879 

65430 

75623 

8 

63 

815924 

24.3 

878547 

18.2 

937376 

42.6 

062624  i  65452 

76604 

7 

54 

816069 

24.3 

878438 

18.2 

937632 

42.5 

062368  165474 

75685 

6 

65 

816216 

24.3 

878328 

18.2 

937887 

42.6 

062113  j  65496 

75566 

6 

56 

816361 

24.3 

878219 

18.2 

938142 

42.5 

4O  K. 

061858  1  65618 

75547 

4 

57 

816607 

24.3 

878109 

18.  3 

938398 

42.  o 

4O  K 

•  061602  i  65540 

76628 

3 

68 

816652 

24.2 

877999 

18.3 

938653 

42.  o 

061347  |!  65562 

75509 

2 

59 

816798 

24.2 

877890 

18.3 

938908 

42.5 

061092  !j  65584 

75490 

1 

60 

816943 

24.2 

877780 

18.3 

939163 

42.6 

060837  ji  65606 

75471 

0 

Cosine. 

Sine. 

~Cotang. 

Tang.   H  N.  cos. 

N.sine. 

i 

49  Degrees. 

62          Log.  Sines  and  Tangents.  (41°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  10 

Cosine. 

D.  10 

Tang. 

D.  10 

Cotang. 

|N.  sine 

N.cos 

0 

9.816943 

n  A  o 

9.877780 

1  Q   O 

9.939163 

An  E 

10.060837 

6660 

75471 

60 

1 

817088 

24*.  2 

877670 

lO.c 

939418 

4Z.D 

060582 

6662 

75462 

59 

2 

817233 

24.2 

877560 

18.  £ 

939673 

42.  { 

ACt   K 

060327 

6665 

75433 

58 

3 

817379 

24.2 

877450 

18.  £ 

1  Q  O 

939928 

4y  .0 

49  K 

060072 

65672 

76414 

57 

4 

817524 

24.2 

877340 

lo.c: 

940183 

4-*  .  <i 

ACt  K. 

059817  65694 

75395 

56 

5 

817668 

24.1 

877230 

18.2 

940438 

<tZ  .0 

Af)   K. 

059562  65716 

75376 

65 

6 

817813 

24.1 

877120 

18.4 

940694 

4!a.O 

AC)  C 

059306  II  65738 

75356 

64 

7 
8 

817958 
818103 

24.1 
24.1 

877010 
876899 

18.4 
18.4 

940949 
941204 

%».i 

42.5 

/IO  K 

059051  65759 
058796  16578 

75337 
75318 

53 
62 

9 

818247 

24.1 

ClA   1 

876789 

18.4 

184. 

941458 

42.  0 
40  p; 

058542!  165803 

75299 

51 

10 

818392 

z4.  1 

876678 

1O.4 
10  t 

941714 

4,*  .  C 

/tO  K. 

058286  6582 

75280 

60 

11 

9.818536 

24.  1 

9.876568 

lo.4 

9.941968 

42  .  U 

/tO  K. 

10.0580321  6584 

75261 

49 

12 

818681 

24.  C 

876457 

18.4 

tO   A 

942223 

42.  O 

/IO  K 

0577771  65869 

75241 

48 

13 

818825 

24.  C 

876347 

lo.4 

942478 

42  ,  o 

/IO  K 

057522 

6589 

76222 

47 

14 

818969 

24.  C 

876236 

18.4 

942733 

42.  Q 

AC»  e 

057267  65913 

75203 

46 

15 

819113 

24.0 

876125 

18.5 

942988 

42  .  O 

/1O  PL 

057012  ||  65935 

75184 

46 

16 

819257 

24.0 

876014 

18.5 

943243 

42.  O 

/IO  R 

0567571  65956 

76165 

44 

17 

819401 

24.0 

C»A  n 

875904 

18.5 

1  o  c 

943498 

42.  0 
4"}  Ft 

056502  i  :  65978 

75146 

43 

18 

819545 

^4.  0 

875793 

io.O 

943762 

**.£  .  C 

AC)  X 

0562481  66000 

75126 

42 

19 

819689 

23.9 

876682 

18.5 

944007 

42.  0 
/IO  K. 

055993  66022 

75107 

41 

20 

819832 

23.9 

875571 

18.5 

944262 

42.  0 

/IO  R 

055738  66044 

75088 

40 

21 

22 

9.819976 
820120 

23.9 
23.9 

9  875469 
875348 

18.6 

18.5 

9.944517 
944771 

42.  0 

42.5 

/IO  A 

10.  055483'!  66066 
055229  166088 

76069 
75050 

39 
38 

23 

820263 

23.9 

876237 

18.5 

945026 

45J.4 

An  A 

054974  j  66  109 

75030 

37 

24 

82040o 

23.9 

875126 

18.5 

945281 

4IZ.4 

Af)   A 

054719  J66131 

75011 

36 

25 

820550 

23.9 

875014 

18.6 

945535 

4X.  4 

054465  j'  66153 

74992 

35 

26 

820693 

23.8 

874903 

18.6 

945790 

42.4 

ACt  A 

054210!  66175 

74973 

34 

27 

820836 

23.8 

874791 

18.  6 

946045 

4ii  .4 

4O  A 

053955  i  66197 

74953 

33 

28 

820979 

23.8 

874680 

18  .6 
10  a 

946299 

ii.4 

4O  A 

053701  66218 

74934 

32 

29 

821122 

23.8 

874668 

lo.o 

946564 

y.4 

Act  A 

053446  66240 

74916 

31 

30 

821265 

23.8 

874456 

18.6 

946808 

4ii.4 
IO  A 

053192!  66262 

74896 

30 

31 

9.821407 

23.8 

Oo  U 

9.874344 

18.6 

1  Q  fl 

9.947063 

41^.4 
49  4 

10.  052937  jj  66284 

74876 

29 

32 

821550 

^O.  0 

874232 

lo  .0 

947318 

^x^  •  ^fc 
Act  A 

062682  166306 

74857 

28 

33 

821693 

23.8 

874121 

18.7 

947672 

4^.4 

ACt  A 

052428  6632? 

74838 

27 

34 

821835 

23.7 

874009 

18.7 

947826 

4ii.4 

052174 

66349 

74818 

26 

35 

821977 

23.7 

873896 

18.  7 

948081 

42.4 

051919 

166371 

-4799 

25 

36 

822120 

23.7 

873784 

18.7 

948336 

42.4 

051664 

66393 

-4780 

24 

37 

822262 

23.7 

873672 

18.7 

948590 

42.4 

051410 

66414 

-4760 

23 

38 

822404 

23.7 

873560 

L8.7 

948844 

42.4 

051156 

66436 

~4741 

22 

39 

822546 

23.7 

873448 

L8.7 

949099 

42.4 

ACt  A 

050901 

66458 

74722 

21 

40 

822688 

23.7 

873336 

L8.7 

949363 

4z.4 

ACt  A 

050647 

66480 

74703 

20 

41 

.822830 

23.6 

9.873223 

18.7 

9.949607 

4:2.4 

10.050393 

66501 

4683 

19 

42 

822972 

23.6 

873110 

18.7 

949862 

42.4 

ACt  A 

050138 

66523 

4663 

18 

43 

823114 

23.6 

872998 

L8.8 

80 

950116 

4i6.4 
49  4 

049884'  66545 

4644 

17 

44 

823255 

23.6 

872885 

.0 

950370 

Q.Z  •  ^= 
Act  A 

049630  {66566 

4625 

16 

45 

823397 

23.6 

872772 

18.8 

950625 

4!i.  4 

ACt  A 

049375|  166588 

4606 

15 

46 

823539 

23.6 

872659 

[8.8 

950879 

4ir5.  4 
Act  A 

049121  '66610 

4586 

14 

47 

823680 

23.6 

872547 

18.8 

80 

951133 

4;<J.4 
49  4 

048867  166632 

4567 

13 

48 

823821 

23.  5 

OQ  K 

872434 

.0 

80 

951388 

4^  .  ^r 

42.4 

048612)  66663 

4548 

12 

49 

823963 

<«u  »  D 

872321 

.  O 

951642 

/lr>  /i 

048358  166675 

4522 

11 

50 

824104 

23.5 

872208 

:8.8 

951896 

4i«.4 

048104 

'  66697 

4509 

10 

51 

.824245 

23.5 

9.872095 

:8.8 

9.952150 

t2.4 

0.047850  66718 

4489 

9 

62 

824386 

23.6 

871981 

i8.9 

952405 

12.4 

047695  6674U 

4470 

8 

53 

824627 

23.6 

871868 

.8.9 

952659 

t2.4 

047341 

66762 

4451 

7 

54 

824668 

23.5 

871755 

.8.9 

952913 

t2.4 

ACt   A 

047087 

66783 

4431 

6 

55 

824808 

23.4 

871641 

.8.9 
8fi 

953167 

1-2.4 
o  *-i 

046833 

66805 

4412 

5 

56 

824949 

23.4 

871528 

.9 

953421 

tSffO 
ct  o 

046679 

66827 

4392 

4 

57 

825090 

23.4 

871414 

.8.9 
8ri 

953676 

t2.  o 

O  Q 

046325 

66848 

4373 

3 

58 

825230 

23.4 

871301 

.y 

953929 

eSp*  v 

20 

046071 

66870 

4353 

2 

69 

826371 

23.4 

871187 

8.9 

954183 

.O 

045817 

66891 

4334 

1 

60 

825511 

23.4 

871073 

8.9 

954437 

r2.3 

045563 

66913 

4314 

0 

Cosine. 

Sine. 

Cotang. 

Tang.   ||N.  cos. 

T.sine. 

' 

48  Degrees. 

TABLE  H.     Log.  Sines  and  Tangents.  (42°)'  Natural  Sines.          63 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang.   |N.  sine. 

N.  cos. 

0 

9.825511 

9.871073 

9.954437 

10.045563 

66913 

74314 

60 

1 

825651 

23.4 

870960 

19.0 

954691 

42.3 

046309 

66935 

74296 

69 

2 

825791 

23.3 

870846 

19.0 

954945 

42.3 

045056 

66966 

74276 

58 

3 

825931 

23.3 

870732 

19.0 

956200 

42.3 

044800 

66978 

74256 

57 

4 

826071 

23.3 

870618 

19.0 

955454 

42.3 

044646 

66999 

74237 

56 

5 

826211 

23.3 

870504 

19.0 

955707 

42.3 

044293 

67021 

74217 

55 

6 

826351 

23.3 

870390 

19.0 

955961 

42.3 

044039 

67043 

74198 

54 

7 

826491 

23.3 

870276 

19.0 

956215 

42.3 

043785 

67064 

74178 

63 

8 

826631 

23.3 

870161 

19.0 

956469 

42.3 

043531 

67086 

74159 

52 

9 

826770 

23.3 

870047 

19.0 

956723 

42.3 

043277 

67107 

74139 

51 

10 

826910 

23.2 

869933 

19.1 

966977 

42.3 

043023 

67129 

74120 

60 

11 
12 

9.827049 

827189 

23.2 
23.2 

9.869818 
869704 

19.  1 
19.1 

9.957231 

967485 

42.3 
42.3 

10.042769 
042515 

67151 
67172 

74100 
74080 

49 
48 

13 

827328 

23  .2 

OQ  O 

869589 

19.1 

967739 

42.3 

042261 

67194 

74061 

47 

14 

827467 

Jo  .  Z 

869474 

19.1 

957993 

42.3 

042007 

67215 

74041 

46 

15 

827606 

23.2 

869360 

19.1 

958246 

42.3 

041754 

67237 

74022 

45 

16 

827745 

23.2 

869245 

19.1 

968600 

42.3 

041500 

67258 

74002 

44 

17 

827884 

23.2 

869130 

19.1 

958754 

42.3 

041246 

67280 

73983 

43 

18 

828023 

23.1 

869016 

19.1 

959008 

42.3 

040992 

67301 

73963 

42 

19 

828162 

23.1 

868900 

19.2 

959262 

42.3 

040738 

67323 

73944 

41 

20 

828301 

23.1 

OQ  1 

868785 

19.2 

959516 

42-3 

040484 

67344 

73924 

40 

21 

9.828439 

2o  .  1 

OQ  1 

9.868670 

19.2 

9.959769 

42-3 

10.040231 

67366 

73904 

39 

22 

828578 

ZO.  1 
OQ  1 

868555 

19.2 

960023 

42.3 

039977 

67387 

73885 

38 

23 
24 

828716 
828855 

ZO.  1 

23.1 

868440 

868324 

19.2 
19.2 

960277 
960531 

42.3 
42.3 

039723 
039469 

67409 
67430 

73865 
73846 

37 
36 

25 

828993 

23.0 

868209 

19.2 

960784 

42.3 

039216 

67452 

73826 

35 

26 

829131 

23.0 

868093 

19.2 

961038 

42.3 

038962 

67473 

73806 

34 

27 

829269 

23.0 

867978 

19.2 

961291 

42.3 

038709 

67495 

73787 

33 

28 

829407 

23.0 

867862 

19.3 

961545 

42.3 

038455 

67516 

73767 

32 

29 

829545 

23.0 

OQ  A 

867747 

19.3 

961799 

42.3 

038201 

67538 

73747 

31 

30 

829683 

zo  .U 

OQ  44 

867631 

19.3 

962052 

42.3 

037948 

67559 

73728 

30 

81 

9.829821 

Zo  .  u 

OO  O 

9.867515 

19.3 

9.962306 

42.3 

10.037694 

67580 

73708 

29 

32 

829959 

z^.y 

oo  n 

867399 

19.3 

962560 

42.3 

037440 

67602 

73688 

28 

33 

830097 

zz.y 

867283 

19.3 

962813 

42.3 

037187 

67623 

73669 

27 

34 

830234 

00  Q 

867167 

19.3 

963067 

42.3 

036933 

67645 

73649 

26 

35 

830372 

22  -9 

867051 

19.3 

963320 

42.3 

036680 

67666 

73629 

25 

36 

830509 

22.9 

866935 

19.3 

963574 

42.3 

036426 

67688 

73610 

24 

37 

830646 

on 

866819 

19.4 

963827 

42.3 

036173 

67709 

73590 

23 

38 

830784 

OQ 

866703 

19.4 

964081 

42.3 

035919 

67730 

73570 

22 

39 

830921 

oo  Q 

866586 

19.4 

964335 

42.3 

036665 

67752 

73551 

21 

40 

831058 

JKt'O 

866470 

19.4 

964588 

42.3 

035412 

67773 

73531 

20 

41 

9.831195 

0*8 

9  ,866353 

19.4 

9.964842 

42.2 

10.035158 

67795 

73511 

19 

42 

831332 

98 

866237 

19.4 

965095 

42.2 

034905 

67816 

73491 

18 

43 

831469 

rtO  tt 

866120 

19.4 

965349 

42.2 

034661 

67837 

73472 

17 

44 

831606 

866004 

19.4 

966602 

42.2 

034398 

67859 

73452 

16 

45 

831742 

oo  ft 

866887 

19.6 

965856 

42.2 

034145 

67880 

73432 

15 

46 

831879 

~  ~  *  o 
oo  ft 

865770 

19.5 

in  K 

966109 

42.2 

033891 

67901 

73413 

14 

47 

832015 

ZZ  .  o 

865653 

19.0 

in  X 

966362 

42.2 

033638 

67923 

73393 

13 

48 

832162 

no  n 

865536 

19.  0 

966616 

42.2 

033384 

67944 

73373 

12 

49 

832288  **•' 

865419 

19.5 

966869 

42.2 

033131 

67965 

73363 

11 

50 
61 

832425  I*?'' 
9  832661  22-7 

865302 
9.865185 

19.5 
19.5 

1  n  X 

967123 
9.967376 

42.2 
42.2 

032877 
10.032624 

67987 
68008 

73333 
73314 

10 
9 

52 

832697  **'J 

865068 

19.0 

967629 

42.2 

032371 

68029 

73294 

8 

53 

832833  ~rl 

864950 

19.5 

967883 

42.2 

032117 

68051 

73274 

7 

64 

832969  '>~*'' 

864833 

19.6 

968136 

42.2 

031864 

68072 

73254 

6 

65 

833105  |;"-J| 

864716 

19.6 

in  a 

968389 

42.2 

031611 

68093 

73234 

5 

56 

833241  g-  2 

864598 

19.0 
in  a 

968643 

42.2 

031357 

68115 

73215 

4 

57 

833377  22-° 

864481 

19.0 
in  K. 

968896 

42.2 

031104 

68136 

73196 

3 

68 

833512!  22  -6 

864363 

19.  o 
in  d 

969149 

42.2 

030851 

68157 

73175 

2 

69 

833648  I  £j-j! 

864245 

19.  o 
10  £: 

969403 

42.2 

030597 

68179 

73155 

1 

60 

833783 

864127 

ly.o 

969656 

42.2 

030344 

68200 

73135 

0 

Cosine. 

Sine. 

Cotangi 

Tang. 

N.  co?. 

N.sine. 

~I~ 

47  Degrees. 

64          Log.  Sines  and  Tangents.  (43°)  Natural  Sines.     TABLE  II. 

'  |   Sine. 

D.  10"|  Cosine. 

D.  llX 

Tang. 

D.  10"l  Cotang.  |jN.8irie 

N.  cos 

0  9.833783 

no  R 

9.864127 

1Q  R 

9.969656 

49  9 

10.030344  (68200 

73135 

60 

1   833919 

££  ,  O 
99  K 

864010 

iy  .0 
i  Q  ^ 

969909 

4.4  .<4 

4.9  9 

030091  H68221 

73116 

59 

2   834054 

,£*.O 

00  K 

863892 

iy  .0 
i  Q  n 

970162 

4^.  At 

49,  9 

029838  68242 

73096 

58 

3   834189 

A*,  O 
OQ  C 

863774 

iy  .  / 

1Q  7 

970416 

-  1  ~  .  ^ 
49,  1 

029584 

68264 

73076 

57 

4   834325 

A6.O 

QO  K 

863656 

iy  .  / 

1Q  7 

970669 

*±4  .  £ 
49,  9, 

029331 

68285 

73056 

56 

5   834460 

A£,O 
QO  K 

863538 

iy  .  / 
1Q  7 

970922 

ft^,  A 

42  2 

029078 

68306 

73036 

55 

6   834595 

&A  .  O 

863419 

iy  .  i 

971175 

028825 

68327 

73016 

54 

7   834730 

22.5 

863301 

19.7 

971429 

42.2 

028571 

68349 

72996 

63 

8   834865 

22.5 

863183 

19.7 

971682 

42.2 

xo  o 

028318 

68370 

72976 

52 

9   834999 

22.5 

99  A. 

863064 

19.7 

1  Q  *7 

971935 

4ii  .2 
49  9 

028065 

68391 

72957 

51 

10   835134 

2Z.  4 

OO  A 

862946 

iy.  / 

1  Q  O 

972188 

-  1  J  ,  ~ 
49  9 

027812 

68412 

72937 

50 

11  9.835269 

A>*i  .  4 

OO  A 

9.862827 

iy  .0 

1  Q  Q 

9.972441 

4^  .  ^ 
49  9 

10.027559 

68434 

72917 

49 

12   835403 
13   833638 

44*  .  4 

22.4 

OO  A 

862709 
862590 

iy  .0 
19.8 

1Q  O 

972694 

972948 

4^  .  Z 

42.2 

49  9 

027306 
027052 

68455 
68476 

72897 
72877 

48 

47 

14   835672 

22  .4 

862471 

iy  .o 

973201 

T:^  .  £ 
4O  O 

026799 

68497 

72857 

46 

16  1  835807 

22.4 

OO  A 

862353 

19.8 

1  Q  Q 

973454 

2.  2 
AO  O 

026546 

68518 

72837 

46 

16 

835941 

22.  i± 

OO  A 

862234 

iy.  o 

in  o 

973707 

•  i  ^  .  -' 

4.9  9 

026293 

68539 

72817 

44 

17 

836076 

22  .<* 

OO  Q 

862116 

iy  .0 

1  Q  ft 

973960 

T-.-W  .  ^ 

49  9 

026040 

68561 

72797 

43 

18 

836209 

JSt.  o 

OO  Q 

861996 

iy.o 

1Q   Q 

974213 

rr^  .  J. 

49  9 

025787 

68582 

72777 

42 

19 

836343 

£4.  a 

oo  o 

861877 

iy  .0 
in  c 

974466 

'i^  .  ^ 

49  9 

025534 

68603 

72757 

41 

20 

836477 

22  .  o 

OO  Q 

861758 

iy  .  o 

1  Q  Q 

974719 

^•^  .  ^ 
49  9 

025281 

68624 

72737 

40 

21 

.836611 

Sfie.O 

OO  Q 

9.861638 

iy  .y 

1  Q  Q 

9.974973 

4-  -J  ,  w 

4*2  9 

10.025027 

68645 

72717 

39 

22 

836745 

22.  o 

OO  Q 

861519 

iy.y 

in  n 

975226 

tl-^  .  ^ 
49  9 

024774 

68666 

72697 

38 

23 

836878 

22.  o 

OO  Q 

861400 

iy  .y 

1  Q  Q 

975479 

T:-^  .  ^ 

49  9 

024521 

68688 

72677 

37 

24 

837012 

£t,9 

oo  O 

861280 

iy  .y 

10  n 

975732 

4-«  •  ^ 
xo  o 

024268 

68709 

72657 

36 

25 

837146 

22.  2 

861161 

iy  .  y 

976985 

Q£.2£ 

024015 

68730 

72637 

35 

26 

837279 

22.2 

oo  o 

861041 

19.9 

1  Q  n 

976238 

42.2 

yiO  O 

023762 

68761 

72617 

34 

27 

837412 

22.  J4 

860922 

iy.y 

i  n  f\ 

976491 

4w  .  J 

yiO   O 

023609 

68772 

72697 

33 

28 

837546 

22.2 

OO  O 

860802 

iy.y 

1  Q  n 

976744 

4^.^ 

/iO  O 

023256 

68793 

72577 

32 

29 

837679 

22.  2 

OO  O 

860682 

iy.y 

on  f\ 

976997 

4^,;6 

/4O  O 

023003 

68814 

72557 

31 

30 

837812 

22.  .4 

860662 

2\j  .  0 

9fl  n 

977260 

4^.;6 
/to  o 

022750 

68835 

72537 

30 

31 

.  837945 

22.2 

9.860442 

2\)  .  U 
on  A 

9.977503 

4^.;^ 
/iO  O 

10.022497 

68857 

72517 

29 

32 

838078 

22.2 

860322 

-*U.  U 
9rt  n 

977756 

4-^.^ 
>iO  O 

022244 

68878 

72497 

28 

33 

838211 

22.1 

no  1 

860202 

•*U.  U 
OA  n 

978009 

4^.^ 
AO  O 

021991 

68899 

72477 

27 

34 
35 

838344 
838477 

22.1 
22.1 

860082 
859962 

ZU.U 

20.0 

on  A, 

978262 
978515 

4^.;^ 

42.2 

49  9 

021738 
021485 

68920 
68941 

72457 
72437 

26 

25 

36 

838610 

22. 

859842 

xJU.  U 

on  ri 

978768 

4,*  .  ^ 

/4Q   O 

021232 

68962 

72417 

24 

37 

838742 

22. 

859721 

-*U.  If 
90 

979021 

4^  •  ^ 
/IO  O 

020979 

68983 

72397 

23 

38 

838876 

22. 

859601 

^U. 
on 

979274 

4^  *  ^ 
AO  O 

020726 

69004 

72377 

22 

39 

839007 

22. 

859480 

^U. 
on 

979527 

4^»  ^ 
AO  O 

020473 

69025 

72357 

21 

40 

839140 

22. 

859360 

^U. 

OA 

979780 

4-4  •  ^ 

020220 

69046 

72337 

20 

41 

.839272 

22.0 

9  859239 

i&U. 

OA 

9.980033 

42.2 

10.019967 

6906772317 

19 

42 
43 

839404 
839536 

22.0 
22.0 

859119 
858998 

yu. 
20. 

980286 
980538 

42.2 
42.2 

019714 
019462 

6908872297 
6910972277 

18 
17 

44 

839668 

22.0 

858877 

20.1 

980791 

42.2 

019209 

6913072257 

16 

45 

839800 

22.0 

868766 

20.1 

OA  O 

981044 

42.  1 

/IO  1 

018956 

6915172236 

15 

46 

839932 

22.0 

858635 

2\J  .4 
OA  O 

981297 

4^.  1 

/*O  1 

018703 

6917272216 

14 

47 

840064 

22.0 

858514 

Z\)  .6 
OA  O 

981660 

4^.  1 

ytO  1 

018450 

6919372196 

13 

48 

840196 

21.9 

858393 

!«U.^ 

OA  O 

981803 

4*.  1 

xlO   1 

018197 

69214 

72176 

12 

49 

840328 

21.9 

858272 

2(J.* 

982056 

4^4.  1 

017944 

69235 

72156 

11 

60 

840469 

21.9 

858161 

20.2 

982309 

42.1 

017691 

69256 

72136 

10 

51 

.  840591 

21.9 

9.858029 

20.2 

9.982562 

42.  1 

10.017438 

69277 

72116 

9 

62 

840722 

21.9 

857908 

20.2 

982814 

42.1 
,40  1 

017186 

69298 

72095 

8 

"63 

840854 

21.9 

857786 

20.2 

983067 

42.1 

/IO  1 

016933 

69319 

72076 

7 

54 
56 
56 

840985 
841116 
841247 

21.9 
21.9 
21.8 

867665 
867643 
867422 

20.2 
20.3 
20.3 

983320 
983673 
983826 

4!i.l 
42.1 
42.1 

ACt   1 

016680 
016427 
016174 

69340 
69361 
69382 

72055 
72035 
72015 

6 
6 

4 

67 

841378 

21.8 

857300 

20.3 

984079 

4>6.  1 
/iO  1 

016921 

69403 

71995 

3 

58 

841509 

21.8 

857178 

20.3 

984331 

4^.  1 
/io  i 

015669 

69424 

71974 

2 

69 
60 

841640 
841771 

21.8 
21.8 

857056 
856934 

20.3 
20.3 

984584 
984837 

4^.  i 

42.1 

015416 
015163 

69446 
69466 

71954 
71934 

1 
0 

" 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cosjN.sino. 

/ 

46  Degrees. 

TABLE  H.     Log.  Sines  and  Tangents.  (44°)  Natural  Sines.          65 

' 

Sine. 

D.  10" 

Cosine. 

D.  IV 

Tang. 

D.  W 

Cotang. 

N.  sine 

N.  cos 

0 

9.841771 

O1   Q 

9.856934 

on  o 

9.984837 

AC) 

10.015163 

69466 

71934 

60 

1 

841902 

21.8 

856812 

-ill.  o 

985090 

4-6. 

014910 

69487 

71914 

59 

2 

842033 

21.8 

856690 

20.3 

OA  A 

985343 

42. 

ACI 

014657 

69508 

71894 

58 

3 

842163 

21.8 

856568 

20  .4 

OA  A 

985596 

454. 

014404 

69529 

71873 

57 

4 

842294 

21.7 

01  n 

856446 

20  .  4 

0  A   A 

985848 

42. 

4.9 

014152 

69549 

71853 

66 

6 

842424 

xil  .  7 

O1  rt 

856323 

*f\j  .  4 
on  A 

986101 

4*4  , 
4.9, 

013899 

69570 

71833 

55 

6 

842555 

x4l  .  7 
O1  T 

856201 

•411  .  4 

on  A. 

986354 

4,*. 
4.9 

013646 

69591 

71813 

54 

7 

842685 

54  1  .  7 

O1   T 

856078 

x4u  .  4 
on  A 

986607 

4^. 
4.9, 

013393 

69612 

71792 

63 

8 

842815 

2L  .7 

855956 

**\j  .  4 

986860 

**-«  • 

013140 

69633 

71772 

52 

9 

842946 

21.7 

855833 

20.4 

987112 

42. 

012888 

69654 

71752 

61 

10 

843076 

21.7 

855711 

20.4 

987365 

42. 

012635 

69675 

71732 

50 

11 

9.843206 

21.7 

9.855588 

20.5 

9.987618 

42. 

An 

10.012382 

69696 

71711 

49 

12 

843336 

21.6 

855465 

20.5 

987871 

'**.. 

012129 

69717 

71691 

48 

13 

843466 

21.6 

O1  a 

855342 

20.5 

on  K 

988123 

42. 

4.9 

011877 

69737 

71671 

47 

14 

843595 

54l.b 

O1  c 

855219 

*4U.  0 

90  K 

988376 

4^. 
4.9 

011624 

69758 

71650 

46 

15 

843725 

541.5 
01  ^ 

855096 

*4U.  0 
90  Ft 

988629 

4^  . 
4.9 

011371 

69779 

71630 

45 

16 

843855 

~l  .D 

854973 

~u  .  o 

on  K 

988882 

4.*. 

/1O 

011118 

69800 

71610 

44 

17 

843984 

21.6 

854850 

SfU.O 

989134 

4x6. 

010866 

69821 

71590 

43 

18 

844114 

21.6 

854727 

20.5 

989387 

42. 

010613 

69842 

71569 

42 

19 

844243 

21.5 

854603 

20.6 

989640 

42. 

010360 

69862 

71549 

41 

20 

844372 

21.5 

854480 

20.6 

989893 

42. 

010107 

69883 

71529 

40 

21 

22 

9.844502 
844631 

21.5 
21.5 

01  c 

9.854356 
854233 

20.6 
20.6 
on  a 

9.990145 
990398 

42. 

42. 
4.9 

10.009855 
009602 

69904 
69925 

71508 
71488 

39 

38 

23 

844760 

541.5 

O1   E 

854109 

<#U.  o 
9O  R 

990651 

4^  . 
4.9 

009349 

69946 

71468 

37 

24 

844889 

541  .5 

O1   K. 

853986 

*4u  .  D 
90  fi 

990903 

4^. 
4.9 

009097 

69966 

71447 

36 

25 

845018 

541  .5 

O1   K 

853862 

x£u  .  o 
on  p. 

991156 

4t^  . 
49, 

008844 

69987 

71427 

35 

26 

845147 

J>\  .5 

853738 

x4U  .  O 

991409 

^•*. 

008591 

70008 

71407 

34 

27 

845276 

21.5 

853614 

20.6 

991662 

42. 

008338 

70029 

71386 

33 

28 

845405 

21.4 

853490 

20.7 

991914 

42. 

008086 

70049 

71366 

32 

29 

845533 

21.4 

853366 

20.7 

992167 

42. 

007833 

70070 

71345 

31 

30 

845662 

21.4 

853242 

20.7 

992420 

42. 

ACt 

007680 

70091 

71325 

30 

31 

9.845790 

21.4 

9.853118 

20.7 

9.992672 

4i4. 

10-007328 

70112 

71305 

29 

32 

845919 

21.4 

852994 

20.7 

992925 

42. 

007075 

70132 

71284 

28 

33 

846047 

21.4 

852869 

20.7 

on  rt 

993178 

42. 

/iO 

006822 

70153 

71264 

27 

34 

846176 

21.4 

O1   A 

852745 

20.1 
90  7 

993430 

4ii. 

4.9 

006570 

70174 

71243 

26 

35 

846304 

54i  .4 

852620 

*4U  .  i 

993683 

4^  . 

006317 

70195 

71223 

25 

36 

846432 

21.4 

852496 

20.7 

993936 

42. 

006064 

70215 

71203 

24 

3? 

846560 

21.3 

852371 

20.8 

994189 

42. 

005811 

70236 

71182 

23 

38 

846688 

21.3 

852247 

20.8 

994441 

42. 

005559 

70257 

71162 

22 

39 

846816 

21.3 

852122 

20.8 

994694 

42. 

005306 

70277 

71141 

21 

40 

846944 

21.3 

851997 

20.8 

994947 

42. 

005063 

70298 

71121 

20 

41 

9.847071 

21.3 

9.851872 

20.8 

9.995199 

42. 

10-004801 

70319 

71100 

19 

42 

847199 

21.3 

851747 

20.8 

995452 

42. 

004548 

70339 

71080 

18 

43 

847327 

21.3 

851622 

20.8 

995705 

42. 

004295 

70360 

71059 

17 

44 

847454 

21.3 

851497 

20.8 

995957 

42. 

y<O 

004043 

70381 

71039 

16 

45 

847582 

21.2 

851372 

20.9 

996210 

4^. 

yiO 

003790 

70401 

71019 

15 

46 

847709 

21.2 

851246 

20.9 

996463 

4xi. 

003537 

70422 

70998 

14 

47 

847836 

21.2 

851121 

20.9 

996715 

42. 

003285 

70443 

70978 

13 

48 

847964 

21.2 

850996 

20.9 

996968 

42. 

003032 

70463 

70957 

12 

49 

848091 

21.2 

860870 

20.9 

997221 

42. 

/1O  1 

002779 

70484 

70937 

11 

50 

848218 

21.2 

850745 

20.9 

997473 

4x6.  1 
/to  1 

002527 

70505 

70916 

10 

!  51 

9.848345 

21.2 

9.850619 

20.9 

9.997726 

4.4.  1 

10-002274 

70525 

70896 

9 

52 

848472 

21.2 

850493 

20.9 

997979 

42.1 

002021 

70546 

70875 

8 

/53 

848599 

21.1 

850368 

21.0 

998231 

42.1 

001769 

70567 

70866 

7 

i  54 

848726 

21.1 

860242 

21.0 

998484 

42.1 

001516 

70587 

70834 

6 

55 

848852 

21.1 

850116 

21.0 

998737 

42.1 

>4O   1 

001263 

70608 

70813 

6 

56 

848979 

21.1 

849990 

21.0 

998989 

454.  1 

AO  1 

001011 

70628 

70793 

4 

57 

849106 

21.1 

849864 

21.0 

999242 

4-i.  1 

AD  i 

000758 

70649 

70772 

3 

58 

849232 

21.1 
01  1 

849738 

21.0 
01  n 

999495 

'**>  .  1 
4.9,  1 

000505 

70670 

70752 

2 

59 

849359 

«1  .  1 

849611 

*4l  .  U 
01  n 

999748 

4x4  .  1 

yiO   1 

000253 

70690 

70731 

1 

60 

849485 

21.1 

849485 

21  .0 

10.000000 

454.  * 

000000 

70711 

^0711 

0 

Cosine. 

Sine. 

Cotaiig. 

Tang. 

N.  cos. 

N.sine.l  ' 

46  Degrees. 

66           LOGARITHMS 

TABLE  III. 

LOGARITHMS  OF  NUMBERS. 

FROM  1  TO  200, 

INCLUDING  TWELVE  DECIMAL  PLACES. 

N. 

1 

2 
3 

4 
5 

Log. 

tt. 

Log. 

N. 

Log. 

000000  000000 
301029  995664 
477121  254720 
602059  991328 
698970  004336 

41 
42 
43 
44 
i  45 

612783  856720 
623249  290398 
633468  455580 
643452  676486 
653212  513775 

81 
82 
83 
84 
85 

908485  018879 
913813  852384 
919078  092376 
924279  286062 
929418  925714 

6 
7 
8 
9 
10 

778151  250384 
845098  040014 
903089  986992 
954242  509439 
Same  as  to  1. 

46 

47 
48 
49 
50 

662757  831682 
672097  857926 
681241  237376 
690196  080028 
Same  as  to  5. 

86 
.87 
88 
89 
90 

934498  451244 
939519  252619 
944482  672150 
949390  006645 
Same  as  to  9. 

11 
12 
13 
14 
15 

041392  685158 
079181  246048 
113943  352307 
146128  035678 
176091  259056 

51 
52 
53 
54 
55 

707570  176098 
716003  343635 
724275  869601 
732393  759823 
740362  689494 

91 
92 
93 
94 
95 

959041  392321 
963787  827346 
968482  948554 
973127  853600 
977723  605889 

16 
17 
18 
19 
20 

204119  982656 
230448  921378 
255272  505103 
278753  600953 
Same  as  to  2. 

56 
57 
58 
59 
60 

748188  027006 
755874  855672 
763427  993563 
770852  011642 
Same  as  to  6. 

96 
97 
98 
99 
100 

982271  233040 
986771  734266 
991226  075692 
995635  194598 
Same  as  to  10, 

21 

22 
23 
24 
25 

322219  2947 
342422  680822 
361727  836018 
380211  241712 
397940  008672 

61 
62 
63 
64 
65 

785329  835011 
792391  699498 
799340  549453 
806179  973984 
812913  366643 

101 
102 
103 
104 
105 

004321  373783 
008600  171762 
012837  224705 
017033  339299 
021189  299070 

26 
27 
28 
29 
30 

414973  347971 
431363  764159 
447158  031342 
462397  997899 
Same  as  to  3. 

66 
67 
68 
69 
70 

819543  935542 
826074  802701 
832508  912706 
838849  090737 
Same  as  to  7. 

108 
107 
108 
109 
110 

025305  865265 
029383  777685 
033423  755487 
037426  497941 
game  as  to  11. 

31 
32 
33 
34 
35 

491361  693834 
505149  978320 
518513  939878 
531478  917042 
544068  044350 

71 
72 
73 
74 
75 

851258  348719 
857332  496431 
863322  860120 
8'69231  719731 
875061  263392 

111 
112 
113 
114 
115 

045322  978787 
049218  022670 
053078  443483 
056904  851336 
060697  840354 

36 
37 
38 
39 
40 

556302  500767 
568201  724067 
579783  596617 
591064  607026 
Same  as  to  4. 

76 
77 
78 
79 
80 

880813  592281 
886490  725172 
892094  602690 
897627  091290 
Same  as  to  8. 

116 
117 
118 
119 
120 

064457  989227 
068185  861746 
071882  007306 
076546  961393 
Same  as  to  12. 

OF  NUMBERS.            67 

N. 

Log- 

i  X. 

Log. 

N. 

Log 

121 
122 
123 
124 
125 

082785  370316 
086369  830675 
089905  111439 
093421  685162 
096910  013008 

148 
149 
150 
151 
152 

170261  715395 
173186  268412 
176091  259056 
178976  947293 
181843  587945 

175 
176 
177 
178 
179 

243038  048686 
245512  667814 
247973  266362 
250420  002309 
252853  030980 

126 
127 
128 
129 
130 

100370  545118 
103803  720956 
107209  969648 
110589  710299 
Same  as  to  13. 

153 
154 
155 
156 
157 

184691  430818 
187520  720836 
190331  698170 
193124  588354 
195899  652409 

180 
181 

182 
183 
184 

255272  505103 
257678  674869 
260071  387985 
262451  089730 
264817  823010 

131 
132 
133 
134 
135 

117271  295656 
120573  931206 
123851  640967 
127104  798365 
130333  768495 

158 
159 
160 
161 
162 

198657  086954 
201397  124320 
204119  982656 
206825  876032 
209515  014643 

185 
186 

187 
188 
189 

267171  728403 
269512  944218 
271841  606536 
274157  849264 
276461  804173 

136 
137 
138 
139 
140 

133538  908370 
136720  567156 
139879  086401 
143014  800254 
146128  035678 

163 
164 
165 
166 
167 

212187  604404 
214843  848048 
217483  944214 
220108  088040 
222716  471148 

190 
191 
192 
193 
194 

278763  600953 
281033  367248 
283301  228704 
285557  309008 
287801  729930 

141 
142 
143 
144 
145 

149219  112655 
152288  344383 
155336  037465 
158362  492095 
161368  002235 

168 
169 
170 
171 
172 

225309  281726 
227886  704614 
230448  921378 
232996  110392 
235528  446908 

195 
196 
197 
198 
199 

290034  611362 
292256  071356 
294466  226162 
296665  190262 
298853  076410 

146 
147 

164352  855784 
167317  334748 

173 
174 

238046  103129 
240549  248283 

LOGARITHMS  OF  THE  PRIME  NUMBERS 

FROM  200  TO  1543, 

INCLUDING  TWELVE  DECIMAL  PLACES. 

.  N. 

Log. 

N. 

Log. 

N. 

Log. 

201 
203 
207 
209 
211 

303196  057420 
307496  037913 
315970  345457 
320146  286111 
324282  455298 

277 
281 
283 
293 
307 

442479  769064 
448706  319905 
451786  435524 
466867  620354 
487138  375477 

379 
383 
389 
397 
401 

678639  209968 
683198  773968 
589949  601326 
698790  606763 
603144  372620 

223 
227 
229 
233 
239 

348304  863048 
356025  857193 
359835  482340 
367355  921026 
378397  900948 

311 
313 
317 
331 
337 

492760  389027 
495544  337546 
601059  262218 
619827  993776 
527629  900871 

409 
419 
421 
431 
433 

611723  308007 
622214  022966 
624282  095836 
634477  270161 
636487  896363 

241 
251 
257 
263 
269 

382017  042575 
399673  721481 
409933  123331 
419955  748490 
429752  280002 

347 
349 
353 
359 
367 

540329  474791 
542825  426959 
547774  705388 
555094  448578 
564666  064262 

439 

443 
449 
457 
461 

642424  620242 
646403  726223 
652246  341003 
659916  200070 
663700  925390 

271 

432969  290874 

373 

571708  831809 

463 

665580  991018 

68           LOGARITHMS 

N. 

Log. 

N. 

Log, 

N. 

Log. 

467 

669316  880566 

821 

914343  157119 

TIT!" 

Uo«5c6  895072 

479 

680335  513414 

823 

915399  836212 

1181 

072249  807613 

487 

687f;28  961215 

827 

917505  509553 

1187 

074450  718955 

491 

691081  492123 

829 

918554  530550 

1193 

076640  443670 

499 

698100  645523 

839 

923761  960829 

1201 

0;9543  007385 

603 

701567  985056 

853 

930949  031168 

1213 

083860  800845 

509 

706717  782337 

857 

932980  821923 

1217 

085290  578210 

521 

716837  723300 

859 

933993  163831 

1223 

087426  458017 

623 

718501  688867 

863 

936010  795716 

1229 

089551  882866 

641 

733197  265107 

877 

942999  593356 

1231 

090258  052912 

547 

737987  326333 

881 

944975  908412 

1237 

092369  699609 

657 

745855  195174 

883 

945960  703578 

1249 

096562  438356 

663 

750508  394851 

887 

947923  619832 

1269 

100026  729204 

669 

755112  266395 

907 

957607  287060 

1277 

106190  896808 

671 

756636  108246 

911 

959518  376973 

1279 

106870  542460 

677 

761176  813166 

919 

963315  511386 

1283 

108226  656362 

587 

768638  101248 

929 

968015  713994 

1-289 

110252  917337 

693 

773064  693364 

937 

971739  690888 

1291 

110926  242517 

699 

777426  822389 

941 

973589  623427 

1297 

112939  986066 

601 

778874  472002 

947 

976349  979003 

1301 

114277  2y6o40 

607 

783138  6910/6 

953 

979092  900638 

1303 

114944  415712 

613 

787460  474518 

967 

985426  474083 

1307 

116275  587564 

617 

790285  164033 

971 

987219  229908 

1319 

120244  795568 

619 

791690  649020 

977 

989894  663719 

1321 

120902  817604 

631 

800029  359244 

983 

992553  617832 

1327 

122870  922849 

641 

806858  029519 

991 

996073  654485 

1361 

133858  125188 

643 

808210  972924 

997 

998695  158312 

1367 

135768  514554 

647 

810904  280669 

1009 

003891  166237 

1373 

137670  537223 

653 

814913  181275 

1013 

005609  445360 

1381 

140193  678544 

659 

818885  414594 

1019 

008174  184006 

1399 

145817  714122 

661 

810201  459486 

1021 

009025  742087 

1409 

148910  994096 

673 

828015  064224 

1031 

013258  665284 

1423 

153204  896557 

677 

830588  668685 

1033 

014100  321520 

1427 

154424  012366 

683 

834420  703682 

1039 

016615  647657 

1429 

155032  228774 

691 

839478  047374 

1049 

020775  488194 

1433 

166246  402184 

701 

845718  017967 

1051 

021602  716028 

1439 

158060  793919 

709 

850646  235183 

1061 

025715  383901 

1447 

160468  531109 

719 

856728  890383 

1063 

026533  264523 

1451 

161667  412427 

727 

861534  410859 

1069 

028977  705209 

1453 

162265  614286 

733 

865103  974742 

1087 

036229  644086 

1459 

164055  291883 

739 

868G44  488395 

1091 

037824  750588 

1471 

167612  672629 

743 

870988  813761 

1093 

038620  161950 

1481 

170555  058512 

751 

855639  937004 

1097 

040206  627675 

1483 

171141  151014 

767 

879095  879500 

1103 

042595  512440 

1487 

172310  968489 

761 

881384  666771 

1109 

044931  546119 

1489 

172894  731332 

769 

885926  339801 

1117 

048053  173116 

1493 

174059  807708 

773 

888179  493918 

1123 

050379  756261 

1499 

175801  632866 

787 

895974  732359 

1129 

052693  941925 

1511 

179264  464329 

797 

901458  321396 

1151 

061075  323630 

1623 

182699  903324 

809 

907948  621612 

1153 

061829  307295 

1531 

184976  190807 

811 

909020  854211 

1163 

065679  714728 

1643 

188365  926053 

OF    NUMBERS. 


69 


AUXILIARY    LOGARITHMS, 


N. 

Log. 

N. 

Log. 

.009 

003891166237  " 

1.0009 

000390689248  > 

.008 

003460532110 

1.0008 

000347296684 

.007 

003029470554 

1.0007 

000303899784 

.  .006 

002598080685 

1,0006 

000260498547 

.005 

002166061756 

A 

1.0005 

000217092970 

>B 

.004 

001733712775 

1.0004 

000173683057 

.003 

001300933020 

1.0003 

000130268804 

.002 

000867721529 

1.0002 

000086850211 

.001 

000434077479  . 

\ 

1.0001 

000043427277 

c 


N. 

.00009 
.00008 
.00007 
.00006 
,00005 
.00004 
.00003 
.00002 
.00001 

Log. 

!   N. 
1.000009 
1  .  000008 
1.000007 
1.000006 
1  .  000005 
1.000004 
1.000003 
1.000002 
1  .  000001 

Log. 

000039083266 
000034740691 
000030398072 
000026055410 
000021712704 
000017371430 
000013028638 
000008685802 
000004342923 

000003908628 
000003474338 
000003040047 
000002605756 
000002171464 
000001737173 
000001302880 
000000868587 
000000434294 

N".         Log. 

1  0000001  000000043429  (n) 
1.00000001  000000004343  (o) 
1.000000001  000000000434  (p) 
1,0000000001  000000000043  (q) 

?n=0.4342944819        log.  —1.637784298. 

By  the  preceding  tables  —  and  the  auxiliaries  A,  JB,  and 
(7,  we  can  find  the  logarithm  of  any  number,  true  to  at  least 
ten  decimal  places. 

But  some  may  prefer  to  use  the  following  direct  formula, 
which  may  be  found  in  any  of  the  standard  works  on  algebra: 

Log.  («+l)=log.«+0.8686889638^_-L. 

The  result  will  be  true  to  twelve  decimal  places,  if  z  be 
over  2000. 

The  log.  of  composite  numbers  can  be  determined  by  the 
combination  of  logarithms,  already  in  the  table,  and  the  prime 
numbers  from  the  formula. 

Thus,  the  number  3083  is  a  prime  number,  find  its  loga- 
rithm. 

We  first  find  the  log.  of  the  number  3082.  By  factoring, 
we  discover  that  this  is  the  product  of  46  into  67. 


70  NUMBERS. 


Log.  46,  1.6627578316 

Log.  67,  1.8260748027 

Log.  3082  3.4888326343 

Log.  3083=3.4888326343+°-8685889638 

6165 


NUMBERS  AND  THEIR  LOGARITHMS, 

OFTEN    USED    IN    COMPUTATIONS. 

Circumference  of  a  circle  to  dia.  1 }  Log. 

Surface  of  a  sphere  to  diameter  IV  =3.14159265  0,4971499 
Area  of  a  circle  to  radius  1  ) 

Area  of  a  circle  to  diameter  1  =  .7853982  — 1.8950899 
Capacity  of  a  sphere  to  diameter  1  =  .5235988—1.7189986 
Capacity  of  a  sphere  to  radius  1  =4.1887902  0.6220886 

Arc  of  any  circle  equal  to  the  radius  =57°29578  1.7581226 
Arc  equal  to  radius  expressed  in  sec.  =206264"8  5.3144251 
Length  of  a  degree,  (radius  unity)=.01745329  — 2.2418773 

12  hours  expressed  in  seconds,      =    43200  4.6354837 

Complement  of  the  same,       =0.00002315  — 5.3645163 

360  degrees  expressed  in  seconds,  =   1296000          6.1 126050 

A  gallon  of  distilled  water,  when  the  temperature  is  62° 
Fahrenheit,  and  Barometer  30  inches,  is  277.  rV&  cubic 
inches. 


^277.274=  16.651 542  nearly. 


4 


277.274 


.775398 


-=18.78925284  ^231=15.198684. 


J  282  =16.792855. 

282." 


._=  18.948708. 


.785398 

The  French  Metre=--3.2808992,  English  feet  linear  mea- 
sure, =39.3707904  inches,  the  length  of  a  pendulum  vi- 
brating seconds. 


TRAVERSE  TABLE.         71 

S 

K  Deg.     ||     1  Deg. 

1M  Deg. 

2  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

1 

1.  00 

0.  01 

1.  00 

0.  02 

1.  00 

0.  03 

1.  00 

0.  03 

2 

2.  00 

0.  02  ; 

2.  00 

0.  03 

2.  00 

0.  05 

2.  00 

0.  07 

3 

3.  00 

0.  03   3.  00 

0.  05 

3.  00 

0.  08 

3.  00 

0.  10 

4 

4.  00 

0.  03  !  4.  00 

0.  07 

4.  00 

0.  10 

4.  00 

0.  14 

5 

5.  00 

0.  04  !  5.  00 

0.  09 

5.  00 

0.  13 

5.  00 

0.  17 

6 

6.  00 

0.  05   6.  90 

0.  10 

6.  00 

0.  16 

6.  00 

0.  21 

7 

7.  00 

0.  06  i  7.  00 

0.  12 

7.  00 

0.  18 

7.  00 

0.  24 

8 

8.  00 

0.  07  i 

8.  00 

0.  14 

8.  00 

0.  21 

7.  99 

0.  28 

9 

9.  00 

0.  08 

9.  00 

0.  16 

9.  00 

0.  24 

8.  99 

0.  31 

10 

10.  00 

0.  09 

10.  00 

0.  17 

10.  00 

0.  26 

9.  99 

0.  35 

11 

11.  00 

0.  10 

11.  00 

0.  19 

11.  00 

0.  28 

10.  99 

0.  38 

12 

12.  00 

o.  10  : 

12.  00 

0.  21 

12.  00 

0.  31 

11.  99 

0.  42. 

13 

13.  00 

0.  11 

13.  00 

0.  23 

13.  00 

0.  34 

12.  99 

0.  45 

14 

14.  00 

0.  12 

14.  00 

0.  24 

14.  00 

0.  37 

13.  99 

0.  49 

15 

15.  00 

0.  13 

15.  00 

0.  26 

14.  99 

0.  39 

14.  99 

0.  52 

16 

16.  00 

0.  14 

16.  00 

0.  28 

15.  99 

0-  42 

15.  99 

0.  56 

17 

17.  00 

0.  15 

17.  00 

0.  30 

16.  99 

0.  45 

16.  99 

0.  59 

18 

18.  00 

0.  16 

18,  00 

0.  31 

17.  99 

0.  47 

17.  99 

0.  63 

19 

19.  00 

0.  17 

19.  00 

0.  33 

18.  99 

0.  50 

18.  99 

0.  66 

20 

20.  00 

0.  17 

20.  00 

0.  35 

19.  99 

0.  52 

19.  99 

0.  70 

21 

21.  00 

0.  18 

21.  00 

0.  37 

20.  99 

0.  55 

20.  99 

0.  73 

22 

22.  00 

0.  19 

22.  00 

0.  38 

21.  99 

0.  58 

21.  99 

0.  77 

23 

23.  00 

0.  20 

23.  00 

0.  40 

22.  99 

0.  60 

22.  99 

0-  80 

24 

24.  00 

0.  21 

24.  00 

0.  42 

23.  99 

0.  63 

23.  99 

0.  84 

25 

25.  00 

0.  22 

25.  00 

0.  44 

24.  99 

0.  65 

24.  98 

0.  87 

26 

26.  00 

0.  23 

26.  00 

0.  45 

25.  99 

0.  68 

25.  98 

0.  91 

27 

27.  00 

0.  24 

27.  00 

0.  47 

26.  99 

0.  71 

26.  98 

0.  94 

28 

28.  00 

0.  24 

28.  00 

0.  49 

27.  99 

0.  73 

27.  98 

0.  98 

29 

29.  00 

0.  25 

29.  00 

0.  51 

28.  99 

0.  76 

28.  98 

1.  01 

30 

30.  00 

0.  26 

30.  00 

0.  52 

29.  99 

0.  79 

29.  98 

1.  05 

35 

35.  00 

0.  31 

34.  99 

0.  61 

34.  99 

0.  92 

34.  98 

1.  22 

40 

40.  00 

0.  35 

39.  99 

0.  70 

39.  99 

1.  05 

39.  98 

1.  40 

45 

45.  00 

0.  39 

44.  90 

0.  79 

44.  99 

1.  18 

44.  97 

1.  57 

50 

50.  00 

0.  44 

49.  99 

0.  87 

49.  98 

1.  31 

49.  97 

1.  74 

55 

55.  00 

0.  48 

54.  99 

0.  96 

54.  98 

1.  44 

54.  97 

1.  92 

60 

60.  00 

0.  52 

59.  90 

0.  05 

59.  98 

1.  57 

59.  96 

2.  09 

65 

65.  00 

0.  57 

64.  99 

.  13 

64.  98 

1.  70 

64.  96 

2.  27 

70 

70.  00 

0.  61 

69.  99 

.  22 

69.  98 

1.  83 

69.  96 

2.  44 

75 

75.  00 

0.  65 

74.  99 

.  31 

74.  97 

1.  96 

74.  95 

2.  62 

80 

80.  00 

0.  70 

79.  99 

.  40 

79.  97 

2.  09 

79.  95 

2.  79 

85 

85.  00 

0.  74 

84.  99 

.  48 

84.  97 

2.  23  84.  95 

2.  97 

90 

90.  00 

0.  79 

89.  99 

.  57 

89.  97 

2.  36  1  89.  95 

3.  14 

95 

90.  00 

0.  83 

94.  99 

.  66 

94.  97 

2.  49 

94.  94 

3.  32 

100 

100.  00 

0.  87 

99.  98 

.  75 

99.  97 

2.  62 

99.  94 

3.  49 

Dep. 

Lat 

Dep. 

Lat 

Dep. 

Lat 

Dep. 

Lat. 

89^  Deg. 

89  Deg. 

88%  Deg. 

88  Deg. 

72          TRAVERSE  TABLE. 

| 

2^  Deg. 

3  Deg. 

Sy2  Deg. 

4  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1.  00 

0.  04 

1.  00 

0.  05 

1.  00 

0.  06 

1.  00 

0.  07 

2 

2.  00 

0.  09 

2.  00 

0.  10 

2.  00 

0.  12 

2.  00 

0.  14 

3 

3.  00 

0.  13 

3.  00 

0.  16 

2.  99 

0.  18 

2.  99 

0.  21 

4 

4.  00 

0.  17 

3.  99 

0.  21 

3.  99 

0.  24 

3.  99 

0.  28 

5 

5.  00 

0.  22 

4.  99 

0.  26 

4.  99 

0.  31 

4.  99 

0.  35 

6 

6.  99 

0.  26 

5.  99 

0.  31 

5.  99 

0.  37 

5.  99 

0.  42 

7 

6.  99 

0.  31 

6.  99 

0.  37 

6.  99 

0.  43 

6.  98 

0.  49 

8 

7.  99 

0.  35 

7.  99 

0.  42 

7.  99 

0.  49 

7.  98 

0.  56 

9 

8.  99 

0.  39 

8.  99 

0.  47 

8.  98 

0.  55 

8.  98 

0.  63 

10 

9.  99 

0.  44 

9.  99 

0.  52 

9.  98 

0.  61 

9.  98 

0.  70 

11 

10.  99 

0.  48 

10.  98 

0.  58 

10.  98 

0.  67 

10.  97 

0.  77 

12 

11.  99 

0.  52 

11.  98 

0.  63 

11.  98 

0.  73 

11.  97 

0.  84 

13 

12.  99 

0.  57 

12.  98 

0.  68 

12.  99 

0.  79 

12.  97 

0.  91 

14 

13.  99 

0.  61 

13.  98 

0.  73 

13.  97 

0.  85 

13.  97 

0.  98 

15 

14.  99 

0.  65 

14.  98 

0.  79 

14.  97 

0.  92 

14.  96 

1.  05 

16 

15.  99 

0.  70 

15.  98 

0.  84 

15.  97 

0.  98 

15.  96 

.  12 

17 

16.  98 

0.  74 

16.  98 

0.  89 

16.  97 

1.  04 

16.  96 

.  19 

18 

17.98 

0.  79 

17.  98 

0.  94 

17.  97 

1.  10 

17.  96 

.  26 

19 

18.  98 

0.  83 

18.  98 

0.  99 

18.  96 

1.  16 

18.  95 

.  33 

20 

19.  98 

0.  87 

19.  97 

1.  05 

19.  96 

1.  22 

19.  95 

.  40 

21 

20.  98 

0.  92 

20.  97 

1.  10 

20.  96 

1.  28 

20.  95 

.  46 

22 

21.  98 

0.  96 

21.  97 

.  15 

21.  96 

1.  34 

21-  95 

.53 

23 

22.  98 

1.  00 

22.  97 

.  20 

22.  96 

1.  40 

22.  94 

.  60 

24 

23.  98 

1.  05 

23.  97 

.  26 

23.  96 

1.  47 

23.  94 

.  67 

25 

24.  98 

.  09 

24.  97 

.  31 

24.  95 

1.  53 

24.  94 

.  74 

26 

25.  98 

.  13 

25.  96 

.  36 

25.  95 

1.  59 

25-  94 

.  81 

27 

26.  97 

.  18 

26.  96 

.  41 

26.  95 

1.  65 

26.  93 

.  88 

28 

27.  97 

.  22 

27.  96 

.  47 

27.  95 

1.  71 

27.  93 

.  95 

29 

28.97 

.  26 

28.  96 

.52 

28.  95 

1.  77 

28-  93 

2.  02 

30 

29.  97 

.  31 

29.  96 

.  57 

29.  94 

1.  83 

29.  93 

2.  09 

35 

34.  97 

.  53 

34.  95 

.  83 

34.  93 

2.  14 

34-  91 

2.  44 

40 

39.  96 

.  75 

39.  95 

2.  09 

39.  93 

2.  44 

39.  90  2.  79 

45 

44.96 

1.  96 

44  94 

2.  36 

44.  92 

2.  75 

44-  89 

3.  14 

50 

49.  95 

2.  18 

49.  93 

2.  62 

49.  91 

3.  05 

49-  88 

3.  49 

55 

54.  95 

2.  40 

54.  92 

2.  88 

54.  90 

3.  36 

54-  87 

3.  84 

60 

59.  94 

2.  62 

59.  92 

3.  14 

59.  89 

3.  66 

59.  83 

4.  19 

65 

64.  94 

2.  84 

64.  91 

3.  40 

64.  88 

3.  97 

64.  84 

4.  53 

70 

69.  93 

3.  05 

69.  90 

3.  66 

69.  87 

4.  27 

69.  83 

4.  88 

75 

74.  93 

3.  27 

74.  90 

3.  93 

74.  86 

4.  58 

74.  82 

5.  23 

80 

79.92 

3.  49 

79.89 

4.  19 

79.  85 

4.  88 

79.  81 

5.  58 

85 

84.  92 

3.  71 

84.  88 

4.  45 

84.  84 

5.  19 

84.  79 

5.  93 

90 

89.  91 

3.  93 

89.  98 

4.  71 

89.  83 

5.  49 

89.  78 

6.  28 

95 

94.  91 

4.  14 

94.  87 

4.  97 

94.  82 

5.  80 

94.  77 

6.  63 

100 

99.  91 

4.  36 

99.  86 

5.23 

99.  81 

6.  10 

99.  76 

6.98 

Dep- 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

87^  Deg. 

87  Deg. 

86^  Deg. 

86  Deg. 

TRAVERSE  TABLE.         73 

| 

&A  Deg. 

5  Deg. 

6K  Deg. 

6  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1.  00 

0.  08 

1.  00 

0.  09 

1.  00 

0.  10 

0.  99 

0.  10 

2 

1.  99 

0.  16 

1.99 

0.  17 

1.  99 

0.  19 

1.  99 

0.  21 

3 

2.  99 

0.  24 

2.  99 

0.  26 

2.  99 

0.  29 

2.  98 

0.  31 

4 

3.  99 

0.  31 

3.  98 

0.  35 

3.  98 

0.  38 

3.  98 

0.  41 

5 

4.  98 

0.  39 

4.  98 

0.  44 

4.  98 

0.  48 

4.  97 

0.  52 

6 

5.  98 

0.  47 

5.  98 

0.  52 

5.  97 

0.  58 

5.  97 

0.  63 

7 

6.  98 

0.  55 

6.  97 

0.  61 

6.  97 

0.  67 

6.  96 

0.  73 

8 

7.  98 

0.  63 

7.  97 

0.  70 

7.  96 

0.  76 

7.  96 

0.  84 

9 

8.  97 

0.  71 

8.  97 

0.  78 

8.  96 

0.  86 

8.  95 

0.  94 

10 

9.  97 

0.  78 

9.  96 

0.  87 

9.  95 

0.  96 

9.  95 

1.  05 

11 

10.  97 

0.  86 

10.  96 

0.  96 

10.  95 

1.  05 

10.  94 

1.  15 

12 

11.  96 

0.  94 

11.  95 

.  05 

11.  94 

1.  15 

11.  93 

1.  25 

13 

12.  96 

1.  02 

12.  95 

.  13 

12.  94 

1.  25 

12.  93 

1.  36 

14 

13.  96 

.  10 

13.  95 

.  22 

13.  94 

1.  34 

13.  92 

1.  46 

15 

14.  95 

.  18 

14.  94 

.  31 

14.  93 

1.  44 

14.  92 

1.  57 

16 

15.  95 

.  26 

15.  94 

.  39 

15.  93 

.  53 

15.  91 

1.  67 

17 

16.  95 

.  33 

16.  94 

.  48 

16.  92 

.  63 

16.  91 

1.  78 

18 

17.  94 

.  41 

17.  93 

.  57 

17.  92 

.  73 

17.  90 

1.  88 

19 

18.  94 

.  49 

18.  93 

.  66 

18.  91 

•  82 

18.  90 

1.  99 

20 

19.  94 

.  57 

19.  92 

.  74 

19.  91 

.  92 

19.  89 

2.  09 

21 

20.  94 

.  65 

20.  92 

1.  83 

20.  90 

2.  01 

20.  88 

2.  20 

22 

21.  93 

.  73 

21.  92 

1.  92 

21.  90 

2.  11 

21.  88 

2.  30 

23 

22.  93 

.  80 

22.  91 

2.  00 

22.  89 

2.  20 

22.  87 

2.  40 

24 

23.  93 

.  88 

23.  91 

2.  09 

23.  89 

2.  30 

23.  87 

2.  51 

25 

24.  92 

.  96 

24.  90 

2.  18 

24.  88 

2.  40 

24.  86 

2.  61 

26 

25.  92 

2.  04 

25.  90 

2.  27 

25.  88 

2.  49 

25.  86 

2.  72 

27 

26.  92 

2.  12 

26.  90 

2.  35 

26.  88 

2.  59 

26.  85 

2.  82 

28 

27.  91 

2.  20 

27.  89 

2.  44 

27.  87 

2.  68 

27.  85 

2.  93 

29 

28.  91 

2.  28 

28.  89 

2.  53 

28.  87 

2.  78 

28.  84 

3.  03 

30 

29.  21 

2.  35 

29.  89 

2.61 

29.  86 

2.  88 

29.  84 

3.  14 

35 

34.  89 

2.  75 

34.  87 

3.  05 

34.  84 

3.  35 

34.  81 

3.  66 

40 

39.  88 

3.  14 

39.  85 

3.  49 

39.  82 

3.  83 

39.  78 

4.  18 

45 

44.  86 

3.  53 

44.  83 

3.  92 

44.  79 

4.  31 

44.  75 

4.  70 

50 

49.  85 

3.  92 

49.  81 

4.  36 

49.  77 

4.  79 

49.  73 

5.  23 

55 

54.  85 

4.  08 

54.  79 

4.  79 

54.  75 

5.  27 

54.  70 

5.  75 

60 

59.  84 

4.  45 

59.  77 

5.  23 

59.  72 

5.  75 

59.  67 

6.  27 

65 

64.  82 

4.  82 

64.  75 

5.  67 

64.  70 

6.  23 

64.  64 

6.  79 

70 

69.  81 

5.  19 

69.  73 

6.  10 

69.  68 

6.  71 

69.  62 

7.  32 

75 

74.  79 

5.  65 

74.  71 

6.  54 

74.  65 

7.  19 

74.  59 

7.  84 

80 

79.  78 

5.  93 

79.  70 

6.  97 

79.  63 

7.  67 

76.  56 

8.  36 

85 

84.  77 

6.  30 

84.  68 

7.  41 

84.  61 

8.  15 

84.  53 

8.  88 

90 

89.  75 

6.  67 

89.  66 

7.  84 

89.  59 

8.  63 

89.  51 

9.  41 

95 

94.  74 

7.  04 

94.  64 

8.28 

94.  56 

9.  11 

94.  48 

9.  93 

100 

99.  73 

7.  41 

99.  62 

8.  72 

99.  54 

9.  58 

99.  45 

10.  43 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

85^  Deg. 

85  Deg. 

84^  Deg. 

84  Deg. 

74          TRAVERSE  TABLE. 

| 

6M  Deg. 

7  Deg. 

1Y2  Deg. 

8  Deg. 

| 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  99 

0.  11 

0.  99 

0.  12 

0.  99 

0.  13 

0.  99 

0.  14 

2 

1.  99 

0.  23 

1.  99 

0.  24 

1.  98 

0.26 

1.  98 

0.  28 

3 

2.  98 

0.  34 

2.  98 

0.  37 

2  97 

0.  39 

2.  97 

0.  42 

4 

3.  97 

0.  45 

3.  97 

0.  49 

3.  97 

0.52 

3.  96 

0.  56 

5 

4.  97 

0.  57 

4.  96 

0.  61 

4.  96 

0.65 

4.  95 

0.  70 

6 

5.  96 

0.  68 

5.  96 

0.  73 

5.  95 

0.78 

5.  94 

0.  84 

7 

6.  96 

0.  79 

6.  95 

0.  85 

6.  94 

0.91 

6.  93 

0.  97 

8 

7.  95 

0.  91 

7.  94 

0.  97 

7.  93 

1.04 

7.  92 

.  11 

9 

8.  94 

1.  02 

8.  93 

1.  10 

8.  92 

1.  17 

8.  91 

.  25 

10 

9.  94 

1.  13 

9.  93 

1.  22 

9.  91 

.31 

9.  90 

.  39 

11 

10.  93 

1.  25 

10.  92 

1.  34 

10.  91 

.44 

10.  89 

.  53 

12 

11.  92 

.  36 

11.  91 

1.  46 

11.  90 

.57 

11.  88 

.  67 

13 

12.  92 

.  47 

12.  90 

1.  58 

12.  89 

.70 

12.  87 

.  81 

14 

13.  91 

.  59 

13.  90 

1.  71 

13.  88 

.83 

13.  86 

.  95 

15 

14.  90 

.  70 

14.  89 

1.  83 

14.  87 

.96 

14.  85 

2.  09 

16 

15.  90 

.  81 

15.  88 

1.  95 

15.  86 

2.09 

15.  84 

2.  23 

17 

16.  89 

.  92 

16.  87 

2.  07 

16.  85 

2.22 

16.  83 

2.  37 

18 

17.  88 

2.  04 

17.  87 

2.  19 

17.  85 

2.35 

17.  82 

2.  51 

19 

18.  88 

2.  15 

18.  86 

2.  32 

18.  84 

2.48 

18.  82 

2.  64 

20 

19.  87 

2.  26 

19.  85 

2.  44 

19.  83 

2.61 

19.  81 

2.  78 

21 

20.  87 

2.  38 

20.  84 

2.  56 

20.  82 

2.74 

20.  80 

2.  92 

22 

21.  86 

2.  49 

21.  84 

2.  68 

21.  81 

2.87 

21.  79 

3.  06 

23 

22.  85 

2.  60 

22.  83 

2.  80 

22.  80 

3.00 

22.  78 

3.  20 

24 

23.  85 

2.  72 

23.  82 

2.  92 

23.  79 

3.  13 

23.  77 

3.  34 

25 

24.  84 

2.  83 

24.  81 

3.  05 

24.  79 

3.26 

24.  76 

3.  48 

26 

25.  83 

2.  94 

25.  81 

3.  17 

25.  78 

3.  39 

25.  75 

3.  62 

27 

26.  83 

3.  06 

26.  80 

3.  29 

26.  77 

3.52 

26.  74 

3.  76 

28 

27.  82 

3.  17 

27.  79 

3.41 

27.  76 

3.65 

27.  73 

3.  90 

29 

28.  81 

3.  28 

28.  78 

3.  53 

28.  75 

3.79 

28.  72 

4.  04 

30 

29.  81 

3.  40 

29.  78 

3.  68 

29.  74 

3.92 

29.  71 

4.  18 

35 

34.  78 

3.  96 

34.  74 

4.  27 

34.  70 

4.57 

34.  66 

4.  87 

40 

39.  74 

4.  53 

39.  70 

4.  87 

39.  66 

5.22 

39.  61 

5.  57 

45 

44.71 

5.  09 

44.  67 

5.  48 

44.  62 

5.87 

44.  56 

6.  26 

50 

49.  68 

5.  66 

49.  63 

6.  09 

49.  57 

6.53 

49.  51 

6.  96 

55 

54.  65 

6.  23 

54.  59 

6.  70 

55.  58 

6.70 

54.  46 

7.  65 

60 

59.  61 

6.  79 

59.  55 

7.  31 

59.  55 

7.  31 

59.  42 

8.  35 

65 

64.  58 

7.  36 

64.  52 

7.  92 

64.  52 

7.92 

64.  37 

9  05 

70 

69.  55 

7.  92 

69.  48 

8.  53 

69.  48 

8.53 

69.  32 

9.  74 

75 

74.  52 

8.  49 

74.  44 

9.  14 

74.  44 

9.  14 

74.  27 

10.  44 

80 

79.  49 

9.  06 

79.  40 

9.  75 

79.  40 

9.75 

79.  22 

11.  13 

85 

84.  45 

9.  62 

84.  37 

10.  36 

84.  37 

10.  36 

84.  17 

1  .  83 

90 

89.  42 

10.  19 

89.  33 

10.  97 

89.  33 

10.97 

89.  12 

12.  53 

95 

94.  39 

10-75 

94.  29 

11.  58 

94.  29 

11.58 

94.  08 

13.22 

100 

99.  36 

11.32 

99.25 

12.  19 

99.  25 

12.  19 

99.  03 

13.  92 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

83^  Deg. 

83  Deg. 

82^Deg. 

82  Deg. 

TRAVERSE  TABLE.         75 

i 

8K  Deg. 

9  Deg. 

9  KDeg- 

10  Deg. 

I 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

0.  99 

0.  15 

0.  99 

0.  16 

0.  99 

0.  17 

0.  98 

0.  17 

2 

1.  98 

0.  30 

1.  98 

0.  31 

1.  97 

0.  33 

1.  97 

0.  35 

3 

2.  97 

0.  44 

2.  96 

0.  47 

2.96 

0.  50 

2.  95 

0.  52 

4 

3.  96 

0.  59   3.  95 

0.  63 

3.  95 

0.  66   3.  94 

0.  69 

5 

4.  95 

0.  74 

4.  94 

0.  78 

4.  93 

0.  83 

4.  92 

0.  87 

6 

5.  93 

0.  89 

5.  93 

0.  94 

5.  92 

0.99 

5.  91 

1.  04 

7 

6.  92 

1.  03 

6.  91 

1.  10 

6.  90 

1.  16  1  6.  89 

1.22 

8 

7.  91 

1.  18 

7.  90 

1.  25 

7.  89 

1.32  j  7.88 

1.  39 

9 

8.  90 

1.  33 

8.  89 

1.  41 

8.  88 

1.  49 

8.  86 

1.  56 

10 

9.  89 

1.  48 

9.  88 

1.  56 

9.  86 

1.  65 

9.  85 

1.  74 

11 

10.  88 

1.  63 

10.  86 

1.  72 

10.  85 

1.  82 

10.  83 

1.  91 

12 

11.  87 

1.  77  j  11.  85 

1.  88 

11.  84 

1.  98 

11.  82 

2.  08 

13 

12.  86 

1.  92  12.  84 

2.  03 

12.  82 

2.  15 

12.  80 

2.  26 

14 

13.  85 

2.  07!  13.  83 

2.  19 

13.  81 

2.  31 

13.  79 

2.  43 

15 

14.  84 

2.  22 

14.  82 

2.  35 

14.  79 

2.  48 

14.  77 

2.  60 

16 

15.  82 

2.  36 

15.80 

2.  50 

15.  78 

2.  64 

15.  76 

2.  78 

17 

16.  81 

2.  51 

!  16.  79 

2.  66 

16.  77 

2.  81 

16.  74 

2.  95 

18 

17.  80 

2.  66 

17.  78 

2.  82 

17.  75 

2.  97 

17.  73 

3.  13 

19 

18.  79 

2.  81 

18.77 

2.  97 

18.  74 

3.  14 

18.  71 

3.  30 

20 

19.  78 

2.  96 

19.  75 

3.  13 

19.  73 

3.  30 

19.  70 

3.  47 

21 

20.  77 

3.  10 

20.  74 

3.  29 

20.  71 

3.  47 

20.  68 

3.  65 

22 

21.  76 

3.  25 

21.  73 

3.  44 

21.  70 

3.  63 

21.  67 

3.  82 

23 

22.  75 

3.  40 

22.  72 

3.  60 

22.  68 

3.  80 

22.  65 

3.  99 

24 

23.  74 

3.  55 

23.  70 

3.  75 

23.  67 

3.  96 

23.  64 

4.  17 

25 

24.  73 

3.  70 

24.  69 

3.  91 

24.  66 

4.  13 

24.  62 

4.  34 

26 

25.  71 

3.  84 

25.  68 

4.  07 

25.  64 

4.  29 

25.  61 

4.  51 

27 

26.  70 

3.  99 

26.  67 

4.  22 

26.  63 

4.  46 

26.  59 

4.  69 

28 

27.  69 

4.  14 

27.  66 

4.  38 

27.  62 

4.  62 

27.  57 

4.  86 

29 

28.  68 

4.  29 

28.  64 

4.54 

28.  60 

4.79 

28.  56 

5.  04 

30 

29.  67 

4.43 

29.  63 

4.  69 

29.  59 

4.  95 

29.  54 

5.21 

35 

34.  62 

5.  17 

34.  57 

5.  48 

34.  52 

5.  78 

34.  47 

6.  08 

40 

39.  56 

5.  91 

39.  51 

6.  26 

39.  45 

6.  60 

39.  39 

6.  95 

45 

44.  51 

6.  65 

44.  45 

7.  04 

44.  38 

7.  43 

44.  32 

7.  81 

50 

49.  45 

7.  39 

49.  38 

7.  82 

49.  32 

8.25 

49.  24 

8.  68 

55 

54.  40 

8.  13 

54.  32 

8.  60 

54.  25 

9.  08 

54.  16 

9.  95 

60 

59.  34 

8.  87 

59.  26 

9.  39 

59.  18 

9.  90 

59.  09 

10.  42 

65 

64.  29 

9.  61 

64.  20 

10.  17 

64.  11 

10,73 

64.  01 

11.  29 

70 

69.  23 

10.  35 

69.  14 

10.  95 

69.  04 

11.55 

68.  94 

12.  16 

75 

74.  18 

11.  09 

74.  08 

11.  73 

73.  97 

12.  38 

73.  86 

13.  02 

80 

79.  12 

11.  82 

79.  02 

12.  51 

78.  90 

13.  20 

78.  78 

13.  89 

85 

84.  07 

12.  56 

83.  95 

13.  30 

83.  83 

14.  03 

83.  71 

14.  76 

90 

89.  01 

13.  30 

88.  89 

14.  08 

88.  77 

14.  85 

88.  63 

15.  63 

95 

93.  96 

14  04Ji93.  83 

14.  86 

93.  70 

15.  68 

93.  56 

16.  50 

100 

98.  90 

14.  78  I 

98.  77 

15.  64 

98.  63 

16.  50 

98.  48 

17.  36 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D«p. 

Lat. 

81^  Deg.    [j     81  Deg.     ||    80J^  Deg. 

80  Deg. 

76          TRAVERSE  TABLE. 

9 

10K  Deg. 

11  Deg. 

UK  Deg- 

12  Deg. 

| 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

0.  98 

0.  18 

0.  98 

0.  19 

0.  98 

0.20 

0.  98 

0.  21 

2 

1.  97 

0.  36 

1.  96 

0.  38 

1.  96 

0.40 

1.  96 

0.  42 

3 

2.  95 

0.  55 

2.  94 

0.  57 

2.  94 

0.60 

2.  93 

0.  62 

4 

3.  93 

0.  73 

3.  93 

0.  76 

3.  92 

0.80 

3.  91 

0.  83 

5 

4.  92 

0.  91 

4.  91 

0.  95 

4.  90 

1.00 

4.  89 

1.  04 

6 

5.  90 

1.  09 

5.  89 

.  14 

5.  88 

1.20 

5.  87 

1.  25 

7 

6.  88 

1.  28 

6.  87 

.  34 

6.  86 

1.40 

6.  85 

1.  46  j 

8 

7.  87 

1.  46 

7.  85 

.  53 

7.  84 

1.59 

7.  83 

1.  66 

9 

8.  85 

1.  64 

8.  83 

.  72 

8.  82 

1.79 

8.  80 

1.  87 

10 

9.  83 

1.  82 

9.  82 

.  91 

9.  80 

1.99 

9.  78 

2.  08 

11 

10.  82 

2.  00 

10.  80 

2.  10 

10.  78 

2.  19 

10.  76 

2.  29 

12 

11.  80 

2.  19 

11.  78 

2.  29 

11.  76 

2.39 

11.  74 

2.  49 

13 

12.  78 

2.  37 

12.76 

2.  48 

12.  74 

2.59 

12.  72 

2.  70 

14 

13.  77 

2.  55 

13.  74 

2.  67 

13.  72 

2.79 

13.  69 

2.  91 

16 

14.  75 

2.  73 

14.  72 

2.  87 

14.  70 

2.99 

14.  67 

3.  12 

16 

15.  73 

2.  92 

15.  71 

2.  86 

15.  68 

3.  19 

15.  65 

3.  33 

17 

16.  72 

3.  10 

16.  69 

3.  05 

16.  66 

3.39 

16.  63 

3.  53 

18 

17.  70 

3.  28 

17.  67 

3.  24 

17.  64 

3.59 

17.  61 

3.  74 

19 

18.  68 

3.  46 

18.  65 

3.  43 

18.  62 

3.79 

18.  58 

3.  95 

20 

19.  67 

3.  64 

19.  63 

3.  63 

19.  60 

3.99 

19.  56 

4.  16 

21 

20.  65 

3.  83 

20.  61 

3.  82 

20.  58 

4.  13 

20.  54 

4.  37 

22 

21.  63 

4.  01 

21.  60 

4.  01 

21.  56 

4.39 

21.52 

4.  57 

23 

22.  61 

4.  19 

22.  58 

4.  20 

22.  54 

4.59 

22.  50 

4.  78 

24 

23.  60 

4.  37 

23.  56 

4.  39 

23.  52 

4.78 

23.  48 

4.  99 

25 

24.  58 

4.  56 

24.  54 

4.  58 

24.  50 

4.98 

24.  45 

5.  20 

26 

25.  56 

4.  74 

25.  52 

4.  77 

25.48 

5.  18 

25.  43 

5.  41 

27 

26.  55 

4.  92 

26.  50 

4.  96 

26.  46 

5.38 

26.  41 

6.  61 

28 

27.  53 

5.  10 

27.  49 

5.  15 

27.  44 

5.58 

27.  39 

5.  82 

29 

28.  51 

5.  28 

28.  47 

5.  34 

28.  42 

5.78 

28.  37 

6.  03 

30 

29.  50 

5.  47 

29.  45 

5.  72 

29.  40 

5.98 

29.  34 

6.  24 

35 

34.  41 

6.  38 

34.  36 

6.  68 

34.  30 

6.98 

34.  24 

7.  28 

40 

39.  33 

7.  29 

39.  27 

7.  63 

39.  20 

7.97 

39.  13 

8.  32 

45 

44.  25 

8.  20 

44.  17 

8.  59 

44.  10 

8.97 

44.  02 

9.  36 

50 

49.  16 

9.  11 

49.  08 

9.  54 

49.  00 

9.97 

48.  91 

10.  40 

55 

54.  08 

10.02 

53.  99 

10.  49 

53.  90 

10.97 

53.  80 

11.  44 

60 

59.  00 

10.93 

58.  90 

11.  45 

58.  80 

11.96 

58.  69 

12.  47 

65 

63.  91 

11.  85 

63.  81 

12.  40 

63.  70 

12.96 

63.  58 

13.  51 

70 

68.  83 

12.76 

68.  71 

13.  36 

68.  59 

13.96 

68.  47 

14.  55 

75 

73.  74 

13.67 

73.  62 

14.  31 

73.  49 

14.95 

73.  36 

15.  59 

80 

78.  66 

14.  58 

78.  53 

15.  26 

78.  39 

15.95 

78.  25 

16.  63 

85 

83.  58 

15.  49 

83.  44 

16.  22 

83.  29 

16.95 

83.  14 

17.  67 

90 

88.  49 

16.  40 

88.  35 

17.  17 

88.  19 

17.94 

88.  03 

18.  71 

95 

93.  41 

17.  31 

93.  25 

18.  13 

93.  09 

18.94 

92.  92 

19.  75 

100 

98.  33 

18.  22 

98.  16 

19.  08 

97.  99 

19.94 

97.  81 

20.  79 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

79J^  Deg. 

79  Deg. 

78^  Deg. 

78  Deg. 

TRAVERSE  TABLE.         77 

1 

12^  Deg. 

13  Deg. 

13K  Deg. 

14  Deg. 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  98 

0.  22 

0.97 

0.23 

0.  97 

0.  23 

0.  97 

0.  24 

2 

1.  95 

0.  43 

1.  95 

0.  46 

1.  95 

0.  47 

1.94 

0.  48 

3 

2.  93 

0.  65 

2.  92 

0.  67 

2.  92 

0.  70 

2.  91 

0.  73 

4 

3.  91 

0.  87 

3.  90 

0.  90 

3.  89 

0.  93 

3.  88 

0.  97 

5 

4.  88 

1.  08 

4.  87 

1.  12 

4.  86 

1.  17 

4.  85 

1.  21 

6 

5.  86 

1.  30 

5.  85 

1.  35 

5.  83 

1.  40 

5.  82 

1.  45 

7 

6.  83 

1.  52 

6.  82 

1.  57 

6.  81 

1.  63 

6.  79 

1.  69 

8 

7.  81 

1.  73 

7.  80 

1.  80 

7.  78 

1.  87 

7.  76 

1.  94 

9 

8.  79 

1.  95 

8.  77 

2.  02 

8.  75 

2.  10 

8.  73 

2.  18 

10 

9.  76 

2.  16 

9.  74 

2.  25 

9.  72 

2.  33 

9  70 

2.  42 

11 

10.  74 

2.  38 

10.  72 

2.  47 

10.  70 

2.  57 

10.  67 

2.  66 

12 

11.  72 

2.  60 

11.  69 

2.  70 

11.  67 

2.  80 

11.  64 

2.  90 

13 

12.  69 

2.  81 

12.  67 

2.  92 

12.  64 

3.  03 

12.  61 

3.  15 

14 

13.  67 

3.  03 

13.  64 

3.  15 

13.  61 

3.  27 

13.58 

3.  39 

15 

14.  64 

3.  25 

14.  62 

3.  37 

14.  59 

3.  50 

14.  55 

3.  63 

16 

15.  62 

3.  46 

15.  59 

3.  60 

15.  56 

3.  74 

15.52 

3.  87 

17 

16.  60 

3.  68 

16.  57 

3.  82 

16.  53 

3.  97 

16.  50 

4.  11 

18 

17.  57 

3.  90 

17.  54 

4.  05 

17.  50 

4.  20 

17.  47 

4.  35 

19 

18.  55 

4.  11 

18.  51 

4.  27 

18.  48 

4.  44 

18.  44 

4.  60 

20 

19.  53 

4.  33 

19.  49 

4.  50 

19.  45 

4.  67 

19.  41 

4.  84 

21 

20.  50 

4.  55 

20.  46 

4.  72 

20.  42 

4.  90 

20.  38 

5.  08 

22 

21.  48 

4.  76 

21.  44 

4.  95 

21.  39 

5.  14 

21.  35 

5.  32 

23 

22.  45 

4.  98 

22.  41 

5.  17 

22.  36 

5.  37 

22.  32 

5.  56 

24 

23.  43 

5.  19 

23.  38 

5.  40 

23.  34 

5.  60 

23.  29 

5.  81 

25 

24.  41 

5.  41 

24.  36 

5.  62 

24.  31 

5.  84 

24.  26 

6.  05 

26 

25.  38 

5.  63 

25.  33 

5.  85 

25.  28 

6.  07 

25.  23 

6  29 

27 

26.  36 

5.  84 

26.  31 

6.  07 

26.  25 

6.  30 

26.  20 

6.  53 

28 

27.  34 

6.  06 

27.  28 

6.  30 

27.  23 

6.  54 

27.  17 

6.  77 

29 

28.  31 

6.28 

28.  26 

6.52 

28.  20 

6.  77 

28.  14 

7.  02 

30 

29.  29 

6.  49 

29.  23 

6.  75 

29.  17 

7.  00 

29.  11 

7.  26 

35 

34.  17 

7.  58 

34.  10 

7.  87 

34.  03 

8.  17 

!  33.  96 

8.  47 

40 

39.  05 

8.  66 

38.  97 

9.  00 

38.  89 

9.  34 

|38.  81 

9.  68 

45 

43.  93 

9.  74 

43.  85 

10.  12 

43.  76 

10.  51 

43.  66 

10.  89 

50 

48.  81 

10.  82 

48.  72 

11.  25 

48.  62 

11.  67 

48.  51 

12.  10 

55 

53.  70 

11.  90 

53.  59 

12.  37 

53.  48 

12.  84 

53.  37 

13.  31 

60 

58.  58 

12.  99 

58.  46 

13.  50 

58.  34 

14.  01 

58.  22 

14  52 

65 

63.  46 

14.  07 

63.  33 

14.  62 

63.  20 

15.  17 

|63.  07 

15.  72 

70 

68.  34 

15.  15 

68.  21 

15.  75 

68.  07 

16.  34  :67.  92 

16.  93 

75 

73.  22 

16.  23 

73.  08 

16.  87 

72.  93 

17.  50  |  72.  77 

18.  14 

80 

78.  10 

17.  32 

77.  95 

18.  00 

77.  79 

18.  68  1  77.  62 

19.  35 

85 

82.  99 

18.  40 

82.  82 

19.  12 

82.  65 

19.  84  !  82.  48 

20.  56 

90 

87.  87 

19.  48 

87.  69 

20.  25 

87.  51 

21.  01 

87.  33 

21.  77 

95 

92.  75 

20.  56 

92.  57 

21.  37 

92.  38 

22.  18JJ92.  18 

22.  98 

100 

97.  63 

21.  64 

97.  44 

22.  50 

97.  24 

23.  34 

97.  03 

24.  19 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

D<;p. 

Lat. 

TlYz  Deg. 

77  Deg. 

76^  Peg. 

76  Deg. 

78          TRAVERSE  TABLE. 

| 

14^  Deg. 

15  Deg. 

15^  Deg. 

16  Deg. 

i 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

0.  97 

0.  25 

0.  97 

0.  26 

0.  96 

0.27 

0.  96 

0.  28 

2 

1.  94 

0.  50 

1.93 

0.  52 

1.  93 

0.53 

1.  92 

0.  55 

3 

2.  90 

0.  75 

2.  90 

0.  78 

2.  89 

0.80 

2.  88 

0.  83 

4 

3.  87 

1.  00 

3.  86 

1.  04 

3.  85 

1.07 

3.  85 

1.  10 

5 

4.  84 

1.  25 

4.  83 

1.  29 

4.  82 

1.  34 

4.  81 

1.  38 

6 

5.  81 

1.  50 

5.  80 

1.  55 

5.  78 

1.60 

5.  77 

1.  65 

7 

6.  78 

1.  75 

6.  76 

1.81 

6.  75 

1.87 

6.  73 

1.  93 

8 

7.  75 

2.  00 

7.  73 

2.  07 

7.71 

2.  14 

7.  69 

2.  21 

9 

8.  71 

2.  25 

8.  69 

2.  33 

8.  67 

2.41 

8.  65 

2.  48 

10 

9.  68 

2.  50 

9.  66 

2.  59 

9.  64 

2.67 

9.  61 

2.  76 

11 

10.  65 

2.  75 

10.  63 

2.  85 

10.  60 

2.94 

10.  57 

3.  03 

12 

11.  62 

3.  00 

11.  59 

3.  11 

11.  56 

3.21 

11.  54 

3.  31 

13 

12.  59 

3.  25 

12.  56 

3.  36 

12.  53 

3.47 

12.  50 

3.  58 

14 

13.  55 

3.  51 

13.  52 

3.  62 

13.  49 

3.74 

13.  46 

3.  86 

15 

14.  52 

3.  76 

14.49 

3.  88 

14.  45 

4.01 

14.  42 

4.  13 

16 

15.  49 

4.  01 

15.45 

4.  14 

15.  42 

4.28 

15.  38 

4.41 

17 

16.  46 

4.  26 

16.  42 

4.  40 

16.  38 

4.54 

16.  34 

4.  69 

18 

17.  43 

4.  51 

17.  39 

4.  66 

17.  35 

4.81 

17.  30 

4.  96 

19 

18.  39 

4.  76 

18.  35 

4.  92 

18.  31 

5.08 

18.  26 

5.  24 

20 

19.  36 

5.  01 

19.  32 

5.  18 

19.  27 

5.34 

19.  23 

5.  51 

21 

20.  33 

5.  26 

20.  28 

5.  44 

20.  24 

5.61 

20.  19 

5.  79 

22 

21.  30 

5.  51 

21.  25 

5.  69 

21.  20 

5.88 

21.  15 

6.  06 

23 

22.  27 

5.76 

22.  22 

5.  95 

22.  16 

6.15 

22.  11 

6.  34 

24 

23.  24 

6.  01 

23.  18 

6.  21 

23.  13 

6.41 

23.  07 

6.  62 

25 

24.  20 

6.  26 

24.  15 

6.  47 

24.  09 

6.68 

24.  03 

6.  89 

26 

25.  17 

6.  51 

25.  11 

6.  73 

25.  05 

6.95 

24.  99 

7.  17 

27 

26.  14 

6.  76 

26.  08 

6.  99 

26.  02 

7.22 

25.  95 

7.  44 

28 

27.  11 

7.  01 

27.  05 

7.  25 

26.  98 

7.48 

26.  92 

7.  72 

29 

28.  08 

7.26 

28.  01 

7.51 

27.  95 

7.75 

27  88 

7.  99 

30 

29.  04 

7.  51 

28.  98 

7.76 

28.  91 

8.02 

28.  84 

8.  27 

35 

33.  89 

8.76 

33.  81 

9.  06 

33.  73 

9.35 

33.  64 

9.  65 

40 

38.  73 

10.  02 

38.  64 

10.  35 

38.  55 

10.69 

38.  45 

11.  03 

45 

43.  57 

11.  27 

43.  47 

11.  65 

43.  36 

12.03 

43.  26 

12.40 

50 

48.  41 

12.  52 

48.  30 

12.  94 

48.  18 

13.36 

48.  06 

13.  78 

55 

53.  25 

13.  77 

53.  13 

14.  24 

53.  00 

14.70 

52.  87 

15.  16 

60 

58.  09 

15.  02 

57.  96 

15.  53 

57.  82 

16.03 

57.  68 

16.  54 

65 

62.  93 

16.27 

62.  79 

16.  82 

62.  64 

17.37 

62.  48 

17.  92 

70 

67.  77 

17.  53 

67.  61 

18.  12 

67.  45 

18.71 

67.  29 

19.  29 

75 

72.  61 

18  78 

72.  44 

19.  41 

72.  27 

20.04 

72.  09 

20.  67 

80 

77.  45 

20.  03 

77.  27 

20.  71 

77.  09 

21.38 

76.  90 

22.  05 

85 

82.  29 

21.28 

82.  10 

22-  00 

81.  91 

22.72 

81.  71 

23.  43 

90 

87.  13 

22.53 

86.  93 

23.  29 

86.  73 

24.05 

86.  51 

24.  8.1 

95 

91.  97 

23.  79 

91.76 

24.  59 

91.  54 

25.39 

91.  32 

26.  19 

100 

96.  81 

25.  04 

96.  59 

25.  88 

96.  36 

26.72 

96.  13 

27.56 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

75^  Deg. 

75  Deg. 

74}£  Deg. 

74  Deg. 

TRAVERSE  TABLE.         79 

5 

16^  Deg. 

17  Deg. 

17K  Deg. 

18  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

1 

0.  96 

0.  28 

0.  96 

0.  29 

0.  95 

0.  30 

0.  95 

0.  31 

2 

1.  92 

0.  57 

1.91 

0.58 

1.91 

0.  60 

1.  90 

0.62 

3 

2.  88 

0.  85 

2.  87 

0.  88 

2.  86 

0.  90 

2.  85 

0.  93 

4 

3.  84 

1.  14 

3.  83 

1.  17 

3.  81 

1.  20 

3.  80 

1.  24 

5 

4.  79 

1.  42 

4.  78 

1.  46 

4.  77 

1.  50 

4.  76 

1  55 

6 

5.  75 

1.  70 

5.  74 

1.  75 

5.  72 

1.  80 

5.  71 

1.  85 

7 

6.  71 

1.  99 

6.  69 

2.  05 

6.  68 

2.  10 

6.  66 

2.  16 

8 

7.  67 

2.  27 

7.  65 

2.  34 

7.  63 

2.  41 

7.  61 

2.  47 

9 

8.  63 

2.  56 

8.  61 

2.  63 

8.  58 

2.  71 

8.  56 

2.  78 

10 

9.  59 

2.  84 

9.  56 

2.  92 

9.  54 

3.  01 

9  51 

3.  09 

11 

10.  55 

3.  12 

10.  52 

3.  22 

10.  49 

3.  31 

10.  46 

3.  40 

12 

11.  51 

3.  41 

11.  48 

3.  51 

11.  44 

3.  61 

11.  41 

3.71 

13 

12.  46 

3.  69 

12.  43 

3.  80 

12.  40 

3.  91 

12.  36 

4.  02 

14 

13.  42 

3.  98 

13.  39 

4.  09 

13.  35 

4.21 

13.  31 

4.  33 

15 

14.  38 

4.  26 

14.  34 

4.  39 

14.  31 

4.  51 

14.  27 

4.  64 

16 

15.  34 

4.  54 

15.  30 

4.  68 

15.  26 

4.  81 

15.  22 

4.  94 

17 

16.  30 

4.  83 

16.  26 

4.  97 

16.  21 

5.  11 

16.  17 

5.  25 

18 

17.  26 

5.  11 

17.  21 

5.  26 

17.  17 

5.  41 

17.  12 

5.56 

19 

18.  22 

5.  40 

18.  17 

5.  56 

18.  12 

5.  71 

18.  07 

5.  87 

20 

19.  18 

5.  68 

19.  13 

5.  85 

19.  07 

6.  01 

19.  02 

6.  18 

21 

20.  14 

5.  96 

20.  08 

6.  14 

20.  03 

6.  31 

19.  97 

6.  49 

22 

21.  09 

6.  25 

21.  04 

6.  43 

20.  98 

6.  62 

20.  92 

6.  80 

23 

22.  05 

6.  53 

21.  99 

6.  72 

21.  94 

6.  92 

21.  87 

7.  11 

24 

23.  01 

6.  82 

22.  95 

7.  02 

22.  89 

7.  22 

22.  83 

7.  42 

25 

23.  97 

7.  10 

23.  91 

7.  30 

23.  84 

7.  52 

23.  78 

7.  73 

26 

24.  93 

7.  38 

24.  86 

7.  60 

24.  80 

7.  82 

24.  73 

8.  03 

27 

25.  89 

7.  67 

25.  82 

7.  89 

25.  75 

8-  12 

25.  68 

8.  34 

28 

26.  85 

7.  95 

26.  78 

8.  19 

26.  70 

8.  42 

26.  63 

8.  65 

29 

27.  81 

8.  24 

27.  73 

8.  48 

27.  66 

8.  72 

27.  58 

8.  96 

30 

28.  76 

8.  52 

28.  69 

8.  77 

28.  61 

9.  02 

28.  53 

9.  27 

35 

33.  56 

9.  94 

33.  47 

10.  23 

33.  38 

10.  52 

33.  29 

10.  82 

40 

38.  35 

11.  36 

38.  25 

11.  69 

38.  15 

12.  03 

38.  04 

12.  36 

45 

43.  15 

12.  78 

43.  03 

13.  16 

42.  92 

13-  53 

42.  80 

13.  91 

50 

47.  94 

14.  20 

47.  82 

14.  62 

47.  69 

15-  04 

47.  55 

15.  45 

55 

52.  74 

15.  62 

52.  60 

16.  08 

52.  45 

16.  54 

52.  31 

17.  00 

60 

57.  53 

17.  04 

57.  38 

17.  54 

57.  22 

18.  04 

57.  06 

18.  54 

65 

62.  32 

18.  46 

62.  16 

19.  00 

61.  99 

19.  55 

61.  82 

20.  09 

70 

67.  12 

19.  88 

66.  94 

20.  47 

66.  76 

21.  05 

66.  57 

21.  63 

75 

71.  91 

21.  30 

71.  72 

21.  93 

71.  53 

22.  55 

71.  33  23.  18 

80 

76.  71 

22.  72 

76.  50 

23.  39 

76.  30 

24-  06 

76.  08 

24.  72 

85 

81.  50 

24.  14 

81.  29 

24.  85 

81.  07 

25-  56 

80.  84 

26.  27 

90 

86.  29 

25.  56 

86.  07 

26.  31 

85.  83 

27.  06 

85.  60 

27.  81 

95 

91.  09 

26.  98 

90.  85 

27.  78 

90.  60 

28-  57 

90.  35 

29.  36 

100 

95.  88 

28.  40 

95.  63 

29.  24 

95.  37 

30.  07 

95.  11 

30.  90 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

73^  Deg. 

73  Deg. 

72^  Deg. 

72  Deg. 

80          TRAVERSE  TABLE. 

| 

18^  Deg. 

19  Deg. 

19H  »eg. 

20  Deg. 

| 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  95 

0.  32 

0.  95 

0.  33 

0.  94 

0.  33 

0.  94 

0.  34 

2 

1.  90 

0.  63 

1.  89 

0.  65 

1.  89 

0.67 

1.  88 

0.  68 

3 

2.  84 

0.  95 

2.'  84 

0.  98 

2.  83 

1.00 

2.  82 

1.  03 

4 

3.  79 

1.  27 

3.  78 

1.  30 

3.  77 

1.  34 

3.  76 

1.  37 

5 

4.  74 

1.  59 

4.  73 

1.  63 

4.  71 

1.67 

4.  70 

1.  71 

6 

5.  69 

1.  90 

5.  67 

1.  95 

5.  66 

2.00 

5.  64 

2.  05 

7 

6.  64 

2.  22 

6.  62 

2.  28 

6.  60 

2.  34 

6.  58 

2.  39 

8 

7.  59 

2.  54 

7.  56 

2.  60 

7.54 

2.  67 

7.  52 

2.  74 

9 

8.  53 

2.  86 

8.  51 

2.  93 

8.  48 

3.  Cl 

8.  46 

3.  08 

10 

9.  48 

3.  17 

9.  46 

3.  26 

9.  43 

3.  34 

9.  40 

3.  42 

11 

10.  43 

3.  49 

10.  40 

3.  58 

10.  37 

3.67 

10.  34 

3.76 

12 

11.  38 

3.  81 

11.  35 

3.  91 

11.  31 

4.01 

11.  28 

4.  10 

13 

12.  33 

4.  12 

12.  29 

4.  23 

12.  25 

4.  34 

12.  22 

4.  45 

14 

13.  28 

4.  44 

13.  24 

4.  56 

13.  20 

4.67 

13.  16 

4.  79 

15 

14.  22 

4.  76 

14.  18 

4.  88 

14.  14 

5.01 

14.  10 

5.  13 

16 

15.  17 

5.  08 

15.  13 

5.  21 

15.  08 

5.34 

15.  04 

5.47 

17 

16.  12 

5.  39 

16.  07 

5.  53 

16.  02 

5.67 

15.  97 

5.81 

18 

17.  07 

5.  71 

17.  02 

6.  86 

16.  97 

6.  01 

16.  91 

6.  16 

19 

18.  02 

6.  03 

17.  96 

6.  19 

17.  91 

6.34 

17.  85 

6.  50 

20 

18.  97 

6.  35 

18.  91 

6.51 

18.  85 

6.68 

18.  79 

6.  84 

21 

19.  91 

6.  66 

19.  86 

6.  84 

19.  80 

7.01 

19.  73 

7.  18 

22 

20.  86 

6.  98 

20.  80 

7.  16 

20.  74 

7.34 

20.  67 

7.  52 

23 

21.  81 

7.  30 

21.  75 

7.49 

21.  68 

7.68 

21.  61 

7.  87 

24 

22.  76 

7.  62 

22.  69 

7.  81 

22.  62 

8.01 

22.  55 

8.  21 

25 

23.71 

7.  93 

23.  64 

8.  14 

23.  57 

8.34 

23.  49 

8.  55 

26 

24.  66 

8.  25 

24.  58 

8.  46 

24.  51 

8.68 

24.  43 

8.  89 

27 

25.  60 

8.57 

25.  53 

8.  79 

25.  45 

9.01 

25.  37 

-9.  23 

28 

26.  55 

8.  88 

26.  47 

9.  12 

26.  39 

9.34 

26.  31 

9,  58 

29 

27.  50 

9.  20 

27.  42 

9-  44 

27.  34 

9.68 

27.  25 

9.  92 

30 

28.  45 

9.  52 

28.  37 

9.  77 

28.  28 

10.01 

28.  19 

10.  26 

35 

33.  19 

11.  11 

33.  09 

11.  39 

32.  99 

11.68 

32.  89 

11.  97 

40 

37.  93 

12.  69 

37.  82 

13.  02 

37.  71 

13.  35 

37.  59 

13.  68 

45 

42.  67 

14.  28 

42.  55 

14-  65 

42.  42 

15.02 

42.  29 

15.  39 

50 

47.  42 

15.  87 

47.  28 

16.  28 

47.  13 

16.69 

46.  98 

17.  10 

55 

52.  16 

17.  45 

52.  00 

17.  91 

51.  85 

18.  36 

51.  68 

18.  81 

60 

56.  90 

19.  04 

56.  73 

19.  53 

56.  56 

20.03 

56.  38 

20.  52 

65 

61.  64 

20.  62 

61.  46 

21.  16 

61.  27 

21.70 

61.  08 

22.  23 

70 

66.  38 

22.  21 

66.  19 

22.  79 

67.  98 

23.37 

65.  78 

23.  94 

75 

71.  12 

23.  80 

70.  91 

24.  42 

70.  70 

25.  04 

70.  48 

25.  65 

80 

75.  87 

25.  38 

75.  64 

26.  05 

75.  41 

26.70 

75.  18 

27.  36 

85 

80.  61 

26.  97 

80.  37 

27.  67 

80.  12 

28.37 

79.  87 

29.  07 

90 

85.  35 

28.  56 

85.  10 

29.  30 

84.  84 

30.04 

84.  57 

30.  78 

95 

90.  09 

30.  14 

89.  82 

30.  93 

89.  55 

31.71 

89.  27 

32.  49 

100 

94.  83 

31.73 

94.  55 

32.  56 

94.  26 

33.38 

93.  97 

34.  20 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

71K  Deg. 

71  Deg. 

70X  Deg. 

70  Deg. 

TRAVERSE  TABLE.         81 

g 

20%  Deg. 

21  Deg. 

21%  Deg. 

22  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  94 

0.  35 

0.  93 

0.  36 

0.  93 

0.  37 

0.  93 

0.  37 

2 

1.  87 

0.  70 

1.  87 

0.  72 

1.  86 

0.  73 

1.  85 

0.  75 

3 

2.  81 

1.  05 

2.  80 

1.  08 

2.  79 

1.  10 

2.  78 

1.  12 

4 

3.  75 

1.  40 

3.  73 

1.  43 

3.  72 

1.  47 

3.  71 

1.  50 

5 

4.  68 

1.  75 

4.  67 

1.79 

4.  65 

1.  83 

4.  64 

1.  87 

6 

5.  62 

2.  10 

5.  60 

2.  15 

5.  58 

2.  20 

5.  56 

2.  25 

7 

6.  56 

2.  45 

6.  54 

2.  51 

6.  51 

2.  57 

6.  49 

2.  62 

8 

7.  49 

2.  80 

7.  47 

2.  87 

7.  44 

2.  93 

7.  42 

3.  00 

9 

8.  43 

3.  15 

8.  40 

3.  23 

8.  37 

3.  30 

8.  34 

3.  37 

10 

9.  37 

3.  50 

9.  34 

3.  58 

9.  30 

3.  67 

9  27 

3.  75 

11 

10.  30 

3.  85 

10.  27 

3.  94 

10.  23 

4.  03 

10.  20 

4.  12 

12 

11.24 

4.  20 

11.  20 

4.  30 

11.  17 

4.  40 

11.  13 

4.  50 

13 

12.  18 

4.  55 

12.  14 

4.  66 

12.  10 

4.  76 

12.  05 

4.  87 

14 

13.  11 

4.  90 

13.  07 

5.  02 

13.  03 

5.  13 

12.  98 

5.  24 

15 

14.  05 

5.  25 

14.  00 

5.  38 

13.  96 

5.  50 

13.  91 

5.  62 

16 

14.  99 

5.  60 

14.  94 

5.  73 

14.  89 

5.  86 

14.  83 

5.  99 

17 

15.  92 

5.  95 

15.  87 

6.  09 

15.  82 

6.  23 

15.  76 

6.  37 

18 

16.  86 

6.  30 

16.  80 

6.  45 

16.  75 

6.  60 

16.  69 

6.  74 

19 

17.  80 

6.  65 

17.  74 

6.  81 

17.  68 

6.  96 

17.  62 

7.  12 

20 

18.  73 

7.  00 

18.  67 

7.  17 

18.  61 

7.  33 

18.  54 

7.  49 

21 

19.  67 

7.  35 

19.  61 

7.  53 

19.  54 

7.  70 

19.  47 

7.  87 

22 

20.  61 

7.  70 

20.  54 

7.  88 

20.  47 

8-  06 

20.  40 

8.  24 

23 

21.  54 

8.  05 

21.  47 

8.  24 

21.  40 

8-  43 

21.  33 

8.  62 

24 

22.  48 

8.  40 

22.  41 

8.  60 

22-  33 

8-  80 

22.  25 

8.  99 

25 

23.  42 

8.  76 

23.  34 

8.  96 

23-  26 

9.  16 

23.  18 

9.  37 

26 

24.  35 

9.  11 

24.  27 

9.  32 

24-  19 

9.  53 

24.  11 

9.  74 

27 

25.  29 

9.  46 

25.21 

9.  68 

25-  12 

9.  90 

25.  03 

10.  11 

28 

26.  23 

9.  81 

26.  14 

10.  08 

26-  05 

10.  26 

25.  96 

10.  49 

29 

27.  16 

10.  16 

27.  07 

10.  39 

26-  98 

10.  63 

26.  89 

10.  86 

30 

28.  10 

10.  51 

28.  01 

10.  75 

27-  91 

11.  00 

27.  82 

11.  24 

35 

32.78 

12.  26 

32.  68 

12.  54 

32.  56 

12-  83 

32.  45 

13.  11 

40 

37.  47 

14.  01 

37.  34 

14.  33  37.  22 

14-  66 

37.  09 

14.  98 

45 

42.  15 

15.  76 

42.  01 

16.  13  41.  87 

16.  49  |41.  72 

16.  86 

50 

46.  83 

17.  51 

46.  68 

17.  92  :  46.  52 

18-  33  46.  36 

18.  73 

55 

51.  52 

19.  26 

51.  35 

19.  71  |51.  17 

20-  16|i51.  00 

20.  60 

60 

56.  20 

21.  01 

56.  01 

21.  50  55.  83 

21-  99  ;55.  63 

22.  48 

65 

60.  88 

22.  76 

60.  68 

23.  29  60.  48 

23-  82  |;  60.  27 

24.  35 

70 

65.  57 

24.  51 

65.  35 

25.  09  |  65.  13 

25.  66^64.  90 

26.  22 

75 

70.  25 

26.  27 

70.  02 

26.  88i  69.  78 

27.  49|  69.  54 

28.  10 

80 

74.  93 

28.  02 

74.  69 

28.  67  74.  43 

29.  32  74.  17 

29.  97 

85 

79.  62 

29.  77 

79.  35 

30.  46L79.  09 

31.  15  i  78.  81 

31.  84 

90 

84.  30 

31.  52 

84.  02 

32.  25  i  83.  74 

32.  99 

83.  45 

33.  71 

95 

98.  98 

33.  27 

88.  69 

34.  04 

88.  39 

34.  82  ;  88.  08 

35.  59 

100 

93.  67 

35.  02 

93.  36 

35.  84 

i  93.  04 

36.  65 

92.  72 

37.46 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

69%  Dcg. 

69  Deg. 

68%  Deg. 

68  Deg. 

82          TRAVERSE  TABLE. 

1 

22^  Deg. 

23  Deg. 

23^  Deg. 

24  Deg. 

| 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  90 

0.  38 

0.  92 

0.  39 

0.  92 

0.40 

0.  91 

0.  41 

2 

1.  85 

0.  77 

1.  84 

0.  78 

1.  83 

0.80 

1.  83 

0.  81 

3 

2.  77 

1.  15 

2.  76 

1.  17 

2.  75 

1.20 

2.  74 

1.  22 

4 

3.  70 

1.53 

3.  68 

1.  56 

3.  67 

1.59 

3.  65 

1.  63 

5 

4.  62 

1.91 

4.  60 

1.  95 

4.  59 

1.99 

4.57 

2.  03 

6 

5.  54 

2.  30 

5.  52 

2.  34 

5.  50 

2.39 

5.  48 

2.  44 

7 

6.  47 

2.  68 

6.  44 

2.  74 

6.  42 

2.79 

6.  39 

2.  85 

8 

7.  39 

3.  06 

7.  36 

3.  13 

7.  34 

3.  19 

7.  31 

3.25 

9 

8.  31 

3.  44 

8.  28 

3.  52 

8.  25 

3.59 

8.  22 

3.  66 

10 

9.  24 

3.  83 

9.  20 

3.  91 

9.  17 

3.99 

9  14 

4.  07 

11 

10.  16 

4.21 

10.  13 

4.  30 

10.  09 

4.  39 

10.  05 

4.  47 

12 

11.  09 

4.  59 

11.  05 

4.  69 

11.  00 

4.78 

10.  96 

4.  88 

13 

12.  01 

4.  97 

11.  97 

5.  08 

11.  92 

5.  18 

11.  88 

5.29 

14 

12.  93 

5.  36 

12.  89 

5.  47 

12.  84 

5.58 

12.  79 

5.  69 

15 

13.  86 

5.  74 

13.  81 

5.  86 

13.  76 

5.98 

13.  70 

6.  10 

16 

14.  78 

6.  12 

14.  73 

6.  25 

14.  67 

6.38 

14.  62 

6.  51 

17 

15.  71 

6.  51 

15.  65 

6.  64 

15.59 

6.78 

15.  53 

6.  92 

18 

16.  63 

6.  89 

16.  57 

7.  03 

16.  51 

7.18 

16.  44 

7.  32 

19 

17.  55 

7.27 

17.  49 

7.42 

17.  42 

7.58 

17.  36 

7.  73 

20 

18.  48 

7.  65 

18.  41 

7.  81 

18.  34 

7.97 

18.  27 

8.  13 

21 

19.  40 

8.  04 

19.  33 

8.  21 

19.  26 

8.37 

19.  18 

8.  54 

22 

20.  33 

8.  42 

20.  25 

8.  60 

20.  18 

8.77 

20.  10 

8.  95 

23 

21.  25 

8.  80 

21.  17 

8.  99 

21.  09 

9.  17 

21.  01 

9.  35 

24 

22.  17 

9.  18 

22.  09 

9.  38 

22.  01 

9.57 

21.  93 

9.  76 

25 

23.  10 

9.  57 

23.  01 

9.  77 

22.  93 

9.97 

22.  84 

10.  17 

26 

24.  02 

9.  95 

23.  93 

10.  16 

23.  84 

10.  37 

23.  75 

10.  58 

27 

24.  94 

10.  33 

24.  85 

10.  55 

24.  76 

10.77 

24.  67 

10.  98 

28 

25.  87 

10.  72 

25.  77 

10.  94 

25.  68 

11.  16 

25.  58 

11.  39 

29 

26.  79 

11.  10 

26.  69 

11.33 

26.  59 

11.56 

26.  49 

11.  80 

30 

27.72 

11.  48 

27.  62 

11.  52 

27.51 

11.96 

27.41 

12.  20 

35 

32.34 

13.  39 

32.  22 

13.  68 

32.  10 

13.96 

31.  97 

14.  24 

40 

36.  96 

15.  31 

36.  82 

15.  63 

36.  68 

15.95 

36.  54 

16.  27 

45 

41.  57 

17.22 

41.  42 

17.  58 

41.  27 

17.94 

41.  11 

18.  30 

50 

46.  19 

19.  13 

46.  03 

19.  54 

45.  85 

19.94 

45.  68 

20.  34 

55 

50.  81 

21.  05 

50.  63 

21.  49 

50.  44 

21.93 

50.  24 

22.  37 

60 

55.  43 

22.  96 

55.  23 

23.  44 

55.  02 

23.92 

54.  81 

24.  40 

65 

60.  05 

24.  87 

59.  83 

25.  40 

59.  61 

25.92 

59.  38 

26.  44 

70 

64.  67 

26.  79 

64.  44 

27.  35 

64.  19 

27.91 

63.  95 

28.  47 

75 

69.  29 

28.  70 

69.  04 

29.  30 

68.  78 

29.91 

68.  52 

30.  51 

80 

73.  91 

30.  61 

73.  64 

31.  26 

73.  36 

31.90 

73.  08 

32.  54 

85 

78.53 

32.  53 

78.24 

33.  21 

77.  95 

33.89 

77.  65 

34.  57 

90 

83.  15 

34.  44 

82.  85 

35.  17 

82.  54 

35.89 

82.  22 

36.  61 

95 

87.77 

36.  35 

87.  45 

37.  12 

87.  12 

37.88 

86.  79 

38.  64 

100 

92.  39 

38.27 

92.  05 

39.  07 

91.71 

39.87 

91.  35 

40.  67 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

WA  Deg. 

67  Deg. 

66^  Deg. 

66  Deg. 

TRAVERSE  TABLE.         83 

S 

24^  Deg. 

25  Deg. 

25K  Deg. 

26  Deg. 

f 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  91 

0.  41 

0.  91 

0.  42 

0.  90 

0.  43 

0.  90 

0.  44 

2 

1.  82 

0.  83 

1.81 

0.  85 

1.81 

0.  86 

1.  80 

0.  88 

3 

2.  73 

1.  24 

2.  72 

1.  27 

2.71 

1.  29 

2.  70 

1.  32 

4 

3.  64 

1.  66 

3.  63 

1.  69 

3.  61 

1.  72 

3.  60 

1.  75 

5 

4.  55 

2.  07 

4.  53 

2.  11 

4.  51 

2.  15 

4.  49 

2.  19 

6 

5.  46 

2.  49 

5.  44 

2.  54 

5.  42 

2.  58 

5.  39 

2.  63 

7 

6.  37 

2.  90 

6.  34 

2.  96 

6.  32 

3.  01 

6.  29 

3.  07 

8 

7.  28 

3.  32 

7.  25 

3.  38 

7.  22 

3.  44 

7.  19 

3.  51 

9 

8.  19 

3.  73 

8.  16 

3.  80 

8.  12 

3.  87 

8.  09 

3.  95 

10 

9.  10 

4.  15 

9.  06 

4.23 

9.  03 

4.  31 

8.  99 

4.  38 

11 

10,  01 

4.  56 

9.  97 

4.  65 

9.  93 

4.  74 

9.  89 

4.  82 

12 

10.  92 

4.  98 

10.  88 

5.  07 

10.  83 

5.  17 

10.  79 

5.  26 

13 

11.  83 

5.  39 

11.  78 

5.  49 

11.  73 

5.  60 

11.  68 

5.  70 

14 

12.  74 

5.  81 

12.  69 

5.  92 

12.  64 

6.  03 

12.  58 

6.  14 

15 

13.  65 

6.  22 

13.  59 

6.  34 

13.  54 

6.  46 

13.  48 

6.  58 

16 

14.  56 

6.  64 

14.  50 

6.  76 

14.  44 

6.  89 

14.  38 

7.  01 

17 

15.  47 

7.  05 

15.  41 

7.  18 

15.  34 

7.  32 

15.  28 

7.  45 

18 

16.  38 

7.  46 

16.  31 

7.  61 

16.  25 

7.75 

16.  18 

7.  89 

19 

17.  19 

7.  88 

17.  22 

8.  03 

17.  15 

8-  18 

17.  08 

8.  33 

20 

18.  20 

8.  29 

18.  13 

8.  45 

18.  05 

8.  61 

17.  98 

8.  77 

21 

19.  11 

8.  71 

19.  03 

8.  87 

18.  95 

9-  04  ||  18.  87 

9.  21 

22 

20.  02 

9.  12 

19.  94 

9.  30 

19.  86 

9-  47 

19.  77 

9.  64 

23 

20.  93 

9.  54 

20.  85 

9.  72 

20.  76 

9-  90 

20.  67 

10.  08 

24 

21.  84 

9.  95 

21.  75 

10.  14 

21.  66 

10-  33 

21.  57 

10.  52 

25 

22.  75 

10.  37 

22.  66 

10.  57 

22.  56 

10-  76 

22.  47 

10.  96 

26 

23.  66 

10  78 

23.  56 

10.  99 

23.  47 

11.  19 

23.  37 

11.  40 

27 

24.  57 

11.  20 

24.  47 

U.  41 

24-  37 

11.  62 

24.  27 

11.  84 

28 

25.  48 

11.  61 

25.  38 

U.  83 

25-  27 

12.  05 

25.  17 

12.  27 

29 

26.  39 

12.  03 

26.28 

12.  26 

26-  17 

12-  48 

26.  06 

12.  71 

30 

27.  30 

12.  44 

27.  19 

12.  68 

27-  08 

12-  92 

26.  96 

13.  15 

35 

31.  85 

14.  51 

31.  72 

14.  79 

31.  59 

15-  07 

31.  46 

15.  34 

40 

36.  40 

16.  59 

36.  25 

16.  90 

36.  10 

17.  22 

35.  95 

17.  53 

45 

40.  95 

18.  66 

40.  78 

19.  02 

40-  62 

19-  37 

40.  45 

19.  73 

50 

45.  50 

20.  73 

45.  32 

21.  13 

45.  13 

21-  53 

44.  94 

21.  92 

55 

50.  05 

22.  81 

49.  85 

23.  24 

49.  64 

23-  68 

49.  43 

24.  11 

60 

54.  60 

24.  88 

54.  38 

25.  36 

54.  16 

25-  83 

53.  93 

26.  30 

65 

59.  15 

26.  96 

58.  91 

27  47 

58.  67 

27.  98 

58.  42 

28.  49 

70 

63.  70 

29.  03 

63.  44 

29.  58 

63.  18 

30.  14 

62.  92 

30.  69 

75 

68.  25 

31.  10 

67.  97 

31.  70 

67.  69 

32.  29 

67.  41 

32.  88 

80 

72.  80 

33.  18 

72.  50 

33.  81 

72.  21 

34.  44 

71.  90 

35.  07 

85 

77.  35 

35.  25 

77.  04 

35.  92 

76.  72 

36.  59 

76.  40 

37.  26 

90 

81.  90 

37.  32 

81.  57 

38.  04 

81.  23 

38.  75 

80.  89 

39.  45 

95 

86.  45 

39.  40 

86.  10 

40.  15 

85.  75 

40.  90 

85.  39 

41.  65 

100 

91.  00 

41.  47 

90.  63 

42.  26 

90.  26 

43.  05 

89.  88 

43.  84 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D«p. 

Lat. 

65^  Deg. 

65  Deg. 

64^  Deg. 

64  Deg.     1 

84          TRAVERSE  TABLE. 

| 

26^  Deg. 

27  Deg. 

27^  Deg. 

28  Deg. 

i 

Lat.  1   Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

0.  89 

0.  45 

0.  89 

0.  45 

0.  89 

0.46 

0.  88 

0.  47 

2 

1.  79 

0.  89 

1.78 

0.  91 

1.  77 

0.92 

1.77 

0.  94 

3 

2.  68 

1.  34 

2.  67 

1.  36 

2.  66 

1.39 

2.  65 

1.  41 

4 

3.  58 

1.  78 

3.  56 

1.  82 

3.  55 

1.  85 

3.  53 

1.  88 

5 

4.  57 

2.  23 

4.  45 

2.  27 

4.  44 

2.  31 

4.  41 

2.  35 

6 

5.  37 

2.  68 

5.  35 

2.  72 

5-  32 

2.77 

5.  30 

2.  82 

7 

6.  26 

3.  12 

6.  24 

3.  18 

6-  21 

3.23 

6.  18 

3.  29 

8 

7.  16 

3.  47 

7.  13 

3.  63 

7.  10 

3.69 

7.  06 

3.  76 

9 

8.  05 

4.  02 

8.  02 

4.  09 

7-  98 

4.  16 

7.  95 

4.  23 

10 

8.  95 

4.  46 

8.  91 

4.  54 

8-  87 

4.62 

8.83 

4.  69 

11 

9.  84 

4.  91 

9.  80 

4.  99 

9-  76 

5.08 

9  71 

5.  16 

12 

10.  74 

5.  35 

10.  69 

5.  45 

10-  64 

5.54 

10.  60 

5.  63 

13 

11.  63 

5.  80 

11.  58 

5.  90 

11.  53 

6.00 

11.  48 

6.  10 

14 

12.  53 

6.  25 

12.  47 

6.  36 

12-  42 

6.49 

12.  36 

6.  57 

15 

13.  42 

6.  69 

13.  37 

6.  81 

13-  31 

6.93 

13.  24 

7.  04 

16 

14.  32 

7.  14 

14.  26 

7.  26 

14-  19 

7.  39 

14.  13 

7.  51 

17 

15.  21 

7.  59 

15.  15 

7.72 

15-  08 

7.85 

15.  01 

7.  98 

18 

16.  11 

8.  03 

16.  04 

8.  17 

15-  97 

8.31 

15.  89 

8.  45 

19 

17.  00 

8.  48 

16.  93 

8.  63 

16-  85 

8.77 

16.  78 

8.  92 

20 

17.  90 

8.  92 

17.  82 

9.  08 

17-  74 

9.23 

17.  66 

9.  39 

21 

18.  79 

9.  37 

18.  71 

9.  53 

18-  63 

9.70 

18.  54 

9.  86 

22 

19.  69 

9.  82 

19.  60 

9.  99 

19-  51 

10.  16 

19.  42 

10.  33 

23 

20.  58 

10.  26 

20.  49 

10.  44 

20-  40 

10.62 

20.  31 

10.  80 

24 

21.  48 

10.  71 

21.  38 

10.  90 

21-  29 

11.08 

21.  19 

11.  27 

25 

22.  37  11.  15 

22.  28 

11.  35 

22-  18 

11.54 

22.  07 

11.  74 

26 

23.  27 

11.  60 

23.  17 

11.  80 

23-  06 

12.  01 

22.  96 

12.  21 

27 

24.  16 

12.  05 

24.  06 

12.  26 

23-  95 

12.47 

23.  84 

12.  68 

28 

25.  06 

12.  49 

24.  95 

12.  71 

24-  84 

12.93 

24.  72 

13.  15 

29 

25.  95 

12.  94 

25.  84 

13.  17 

25-  72 

13.  39 

25.  61 

13.  61 

30 

26.  85 

13.  39 

26.  73 

13.  62 

26-  61 

13.85 

26.  49 

14.  08 

35 

31.  32 

15.  62 

31.  19 

15.  89 

31-  05 

16.  16 

30.  90 

16.  43 

40 

35.  80 

17.  85 

35.  64 

18.  16 

35-  48 

18.47 

35.  32 

18.  78 

45 

40.  27 

20.  08 

40.  10 

20.  43 

39.  92 

20.78 

39.  73 

21.  13 

50 

44.  75 

22.  31 

44.  55 

22.  70 

44-  35 

23.  09 

44.  15 

23.  47 

55 

49.  22 

24.  54 

49.  01 

24-  97 

48-  79 

25.40 

48.  56 

25.  82 

60 

53.  70 

26.  77 

53.  46 

27.  24 

53-  22 

27.70 

52.  98 

28.  17 

65 

58.  17 

29.  00 

57.  92 

29.  51 

57.  66 

30.  01 

57.  39 

30.  52 

70 

62.  65 

31.  23 

62.  37 

31.  78 

62.  09 

32.32 

61.  81 

32.  86 

75 

67.  12 

33.  46 

66.  83 

34.  05 

66.  53 

34.63 

66.  22 

35.  21 

80 

71.  59 

35.  70 

71.  28 

36.  32 

70.  96 

36.94 

70.  64 

37.  56 

85 

76.  07 

37.  93 

75.  74 

38-  59 

75.  40 

39.25 

75.  05 

39.  91 

90 

80.  54 

40.  16 

80.  19 

40.  86 

79.  83 

41.56 

79.  47 

42.  25 

95 

85.  02 

42.  39 

84.  65 

43.  13 

84.  27 

43.87 

83.  88 

44.  60 

100 

89.  49 

44.  62 

89.  10 

45.  40 

88.  90 

46.17 

88.  29 

46.  95 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

63^  Deg. 

63  Deg. 

62^  Deg. 

62  Deg. 

TRAVERSE  TABLE.         85 

| 

28^  Deg. 

29  Deg. 

29K  Deg. 

30  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

1 

0.88 

0.  48 

0.  87 

0.  48 

0.87 

0.  49 

0.  87 

0.  50 

2 

1.  76 

0.  95 

1.  75 

0.97 

1.  74 

0.  98 

1.  73 

1.  00 

3 

2.  64 

1.  43 

2.  62 

1.  45 

2.  61 

1.48 

2.  60 

1.  50 

4 

3.  52 

1.  91 

3.  50 

1.94 

3.  48 

1.  97 

3.  46 

2.  00 

5 

4.  39 

2.  39 

4.  37 

2.  42 

4.  35 

2.  46 

4.  33 

2.  50 

6 

5.  27 

2.  86 

5.  25 

2.  91 

5.  22 

2.  95 

5.  20 

3.  00 

7 

6.  15 

3.  34 

6.  12 

3.  39 

6.  09 

3.  45 

6.  06 

3.  50 

8 

7.  03 

3.  82 

7.  00 

3.  88 

6.  96 

3.  94 

6.  93 

4.  00 

9 

7.  91 

4.  29 

7.  87 

4.  36 

7.  83 

4.43 

7.  79 

4.  50 

10 

8.  79 

4.  77 

8.  75 

4.  85 

8.  70 

4.  92 

8.  66 

5.  00 

11 

9.  67 

5.  25 

9.  62 

5.  33 

9.  57 

5.  42 

9.  53 

5.  50 

12 

10.  55 

5.  73 

10.  50 

5.  82 

10.  44 

5.  91 

10.  39 

6.  00 

13 

11.  42 

6.  20 

11.  37 

6.  30 

11.  31 

6.  40 

11.  26 

6.  50 

14 

12.  30 

6.  68 

12.  24 

6.  79 

12.  18 

6.  89 

12.  12 

7.  00 

15 

13.  18 

7.  16 

13.  12 

7.  27 

13.  06 

7.  39 

12.  99 

7.  50 

16 

14.  06 

7.  63 

13.  99 

7.  76 

13.  93 

7.  88 

13.  86 

8.  00 

17 

14.  94 

8.  11 

14.  87 

8.  24 

14.  80 

8-  37 

14.  72 

8.  50 

18 

15.  82 

8.  59 

15.  74 

8,  73 

15.  67 

8.  86 

15.  59 

9.  00 

19 

16.  70 

9.  07 

16.  62 

9.  21 

16.  54 

9.  36 

16.  45 

9.  50 

20 

17.  58 

9.  54 

17.  49 

9.  70 

17.  41 

9.  85 

17.  32 

10.  00 

21 

18.  46 

10.  02 

18.  37 

10.  18 

18.  28 

10.  34 

18.  19 

10.  50 

22 

19.  33 

10.  50 

19.  24 

10.  67 

19.  15 

10.  83 

19.  05 

11.  00 

23 

20.21 

10.  97 

20.  12 

11.  15 

20.  02 

11.  33 

19.  92 

11.  50 

24 

21.  09 

11.  45 

20.  99 

11.  64 

20.  89 

11.  82 

20.  78 

12.  00 

25 

21.  97 

11.  93 

21.  87 

12.  12 

21.  76 

12-  31 

21.  65 

12.  50 

26 

22.  85 

12.  41 

22.  74 

12.  60 

22.  63 

12-  80 

22.  52 

13.  00 

27 

23.  73 

12.  88 

23.  61 

13.  09 

23.  50 

13-  30 

23.  38 

13.  50 

28 

24.  61 

13.  36 

24.  49 

13.  57 

24.  37 

13-  79 

24.  25 

14.  00 

29 

25.  49 

13.  84 

25.  36 

14.  06 

25.  24 

14.  28 

25.  11 

14.  50 

30 

26.  36 

14.  31 

26.  24 

14.  54 

26.  11 

14-  77 

25.  98 

15.  00 

35 

30.  76 

16.  70 

30.  61 

16.  97 

30.  46 

17.  23 

30.  31 

17.  50 

40 

35.  15 

19.  09 

34.  98 

19.  39 

34.  81 

19.  70 

34.  64 

20.  00 

45 

39.  55 

21.  47 

39.  36 

21.  82 

39.  17 

22.  16 

38.  97 

22.  50 

50 

43.  94 

23.  86 

43.  73 

24.  24 

43.  52 

24.  62 

43.  30 

25.  00 

55 

48.  33 

26.  24 

48.  10 

26.  66 

47.  87 

27.  08 

47.  63 

27.  50 

60 

52.  73 

28.  63 

52.  48 

29.  09 

52.  22 

29.  55 

51.  96 

30.  00 

65 

57.  12 

31.  02 

56.  85 

31.  51 

56.  57 

32.  01 

56.  29 

32.  50 

70 

61.52 

33.  40 

61.  22 

33.  94 

60.  92 

34.  47 

60.  62 

35.  00 

75 

65.  91 

35.  79 

65.  60 

36.  36 

65.  28 

36.  93 

64.  95 

37.  50 

80 

70.  31 

38.  17 

69.  97 

38.  78 

69.  63 

39.  39 

69.  28 

40.  00 

85 

74.  70 

40.  56 

74.  34 

41.  21 

73.  98 

41.  86 

73.  61 

42.  50 

90 

79.  09 

42.  94 

78.  72 

43.  63 

78.  33 

44.  32 

77.  94 

45.  00 

95 

83.  49 

45.  33 

83.  09 

46.  06 

82.  68 

46.  78 

82.  27 

47.  50 

100 

87.  88 

47.  72 

87.  46 

48.  48 

87.  04 

49.  24 

86.  60 

50.  00 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

61^  Deg. 

61  Deg. 

60K»eg. 

60  Deg. 

86          TRAVERSE  TABLE. 

« 

£ 

30^  Deg. 

31  Deg. 

31^  Deg. 

32  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  86 

0.  51 

0.  86 

0.  51 

0.  85 

0.  52 

0.  85 

0.  53 

2 

1.72 

1.  02 

1.71 

1.  03 

1.  71 

1.  04 

1.  70 

1.  06 

3 

2.  58 

1.  52 

2.  57 

1.  55 

2.  56 

1.57 

2.  54 

1.  59 

4 

3.  45 

2.  03 

3.  43 

2.  06 

3.  41 

2.  09 

3.  39 

2.  12 

5 

4.  31 

2.  54 

4.  29 

2.  58 

4.  26 

2.  61 

4.  24 

2.  65 

6 

5.  17 

3.  05 

5.  14 

3.  09 

5-  12 

3.  13 

5.  09 

3.  18 

7 

6.  03 

3.  55 

6.  00 

3.  61 

5.  97 

3.  66 

5.  94 

3.  71 

8 

6.89 

4.  06 

6.  86 

4.  12 

6.  82 

4.  18 

6.  78 

4.24 

9 

7.  75 

4.  57 

7.  71 

4.  64 

7.  67 

4.  70 

7.  63 

4.  77 

10 

8.  62 

5.  08 

8.  57 

5.  15 

8.53 

5.  22 

8.  48 

5.  30 

11 

9.  48 

5.  58 

9.  43 

5.  67 

9.  38 

5.  75 

9  33 

5.  83 

12 

10.  34 

6.  09 

10.  29 

6.  18 

10-  23 

6.  27 

10.  18 

6.  36 

13 

11.  20 

6.  60 

11.  14 

6.  70 

11.  08 

6.  79 

11.  02 

6.  89 

14 

12.  06 

7.  11 

12.  00 

7.  21 

11.  94 

7.  31 

11.  87 

7.  42 

15 

12.  92 

7.  61 

12.  86 

7-  73 

12.  79 

7.  84 

12.  72 

7.  95 

16 

13.  79 

8.  12 

13.  71 

8-  24 

13.  64 

8.  36 

13.  57 

8.48 

17 

14.  65 

8.  63 

14.  57 

8-  77 

14.  49 

8.  88 

14.  42 

9.  01 

18 

15.  51 

9.  14 

15.  43 

9-  27 

15.  35 

9.  40 

15.  26 

9.  54 

19 

16.  37 

9.  64 

16.  29 

9-  79 

16.  20 

9.  93 

16.  11 

10.  07 

20 

17.  23 

10.  15 

17.  14 

10-  30 

17-  05 

10.  45 

16.  96 

10.  60 

21 

18.  09 

10.  66 

18.  00 

10-  82 

17.  91 

10.  97 

17.  81 

11.  13 

22 

18.  96 

11.  17 

18.  86 

11-  33 

18-  76 

11.  49 

18.  66 

11.  66 

23 

19.  82 

11.  67 

19.  71 

11-  85 

19.  61 

12.  02 

19.  51 

12.  19 

24 

20.  68 

12.  18 

20.  57 

12-  36 

20-  46 

12.  54 

20.  35 

12.  72 

25 

21.  54 

12.  69 

21.  43 

12-  88 

21.  32 

13.  06 

21.  20 

13.  25 

26 

22.  40 

13.  20 

22.  29 

13-  39 

22-  17 

13.  58 

22.  05 

13.  78 

27 

23.  26 

13.  70 

23.  14 

13-  91 

23.  02 

14.  11 

22.  90 

14.  31 

28 

24.  13 

14.  21 

24.  00 

14-  42 

23.  87 

14.  63 

23.  75 

14.  84 

29 

24.  99 

14.  72 

42.  86 

14-  94 

24-  73 

15.  15 

24.59 

15.  37 

30 

25.  85 

15.23 

25.  71 

15-  45 

25-  58 

15.  67 

25.  44 

15.  90 

35 

30.  16 

17.  76 

30.  00 

18-  03 

29.  84 

18.  29 

29.  68 

18.  55 

40 

34.  47 

20  30 

34.  29 

20-  60 

34.  11 

20.  90 

33.  92 

21.  20 

45 

38.  77 

22.  84 

38.  57 

23-  18 

38.  37 

23.  51 

38.  16 

23.  85 

50 

43.  08 

25.  38 

42.  86 

25.  75 

42.  63 

26.  12 

42.  40 

26.  50 

55 

47.  39 

27.  91 

47.  14 

28.  33 

46.  90 

28.  74 

46.  64 

29.  15 

60 

51.  70 

30.  45 

51.  53 

30-  90 

51.  16 

31.  35 

50.  88 

31.  80 

65 

56.  01 

32.  99 

55.  72 

33.  48 

55.  42 

33.  96 

55.  12 

34.  44 

70 

60.  31 

35.  53 

60.  00 

36.  05 

59.  68 

36.  57 

59.  36 

37.  09 

75 

64.  62 

38.  07 

64.  29 

38.  63 

63.  95 

39.  19 

63.  60 

39.  74 

80 

68.  93 

40.  60 

68.  57 

41.  20 

68.  21 

41.  80 

67.  84 

42.  39 

85 

73.  24 

43.  14 

72.  86 

43.  78 

72.  47 

44.  41 

72.  08 

45.  04 

90 

77.  55 

45.  68 

77.  15 

46.  35 

76.  74 

47.  02 

76.  32 

47.  69 

95 

81.  85 

48.  22 

81.  43 

48.  93 

81.  00 

49.  64 

80.  56 

50.  34 

100 

86.  16 

50.  75 

85.  72 

.51.  50 

85.  26 

52.  25 

84.  80 

52.  59 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

59^  Deg. 

59  Deg. 

58^  Deg. 

58  Deg. 

TRAVERSE  TABLE.         87 

| 

32%  Deg. 

33  Deg. 

33%  Deg. 

34  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  84 

0.  54 

0.  84 

0.  54 

0.  83 

0.  55 

0.  83 

0.  56 

2 

1.  69 

1.  07 

1.  68 

1.  09 

1.  67 

1.  10 

1.  66 

1.  12 

3 

2.  53 

1.  61 

2.  52 

1.  63 

2.50 

1.  66 

2.  49 

1.  68 

4 

3.  37 

2.  15 

3.  35 

2.  18 

3.  34 

2.  21 

3.  32 

2.  24 

5 

4.22 

2.  69 

4.  19 

2.  72 

4.  17 

2.  76 

4.  15 

2.  80 

6 

5.  06 

3.  22 

5.  03 

3.  27 

5.  00 

3.  31 

4.  97 

3.  36 

7 

5.  90 

3.  76 

5.  87 

3.  81 

5.  84 

3.  86 

5.  80 

3.  91 

8 

6.  75 

4.  30 

6.  71 

4.  36 

6.  67 

4.  42 

6.  63 

4.  47 

9 

7.59 

4.  84 

7.  55 

4.  90 

7.  50 

4.  97 

7.  46 

5.  03 

10 

8.  43 

5.  37 

8.  39 

5.  45 

8.  34 

5.  52 

8.29 

5.59 

11 

9.  28 

5.  91 

9.  23 

5.  99 

9.  17 

6.  07 

9.  12 

6.  15 

12 

10.  12 

6.45 

10.  06 

6.  54 

10.  01 

6.  62 

9.  95 

6.  71 

13 

10.  96 

6.  98 

10.  90 

7.  08 

10.  84 

7.  18 

10.  78 

7.  27 

14 

11.  81 

7.  52 

11.  74 

7.  62 

11.  67 

7.  73 

11.  61 

7.  83 

15 

12.  65 

8.  06 

12.  58 

8.  17 

12.  51 

8-  28 

12.  44 

8.  39 

16 

13.  49 

8.  60 

13.  42 

8.  71 

13.  34 

8.  83 

13.  26 

8.  95 

17 

14.  34 

9.  13 

14.  26 

9.  26 

14.  18 

9.  38 

14.  09 

9.51 

18 

15.  18 

9.  67 

15.  10 

9.  80 

15.  01 

9.  93 

14.  92 

10.  07 

19 

16.  02 

10.21 

15.  93 

10.  35 

15.  84 

10.  49 

15.  75 

10.  62 

20 

16.  87 

10.  75 

16.  77 

10.  89 

16.  68 

11.  04 

16.  58 

11.  18 

21 

17.  71 

11.28 

17.  61 

11.  44 

17.  51 

11.  59 

17.  41 

11.  74 

22 

18.  55 

11.  82 

18.  45 

11.  98 

18.  35 

12-  14 

18.  24 

12.  30 

23 

19.  40 

12.  36 

19.  29 

12.  53 

19.  18 

12-  69 

19.  07 

12.86 

24 

20.  28 

12.  90 

20.  13 

13.  07 

20.  01 

13-25 

19.  90 

13.  42 

25 

21.  08 

13.  43 

20.  97 

13.  62 

20.  85 

13.  80 

20.  73 

13.  98 

26 

21.  93 

13.  97 

21.  81 

14.  16 

21.  68 

14.  35 

21.  55 

14.  54 

27 

22.  77 

14.  51 

22.  64 

14.  71 

22.  51 

14-  90 

22.  38 

15.  10 

28 

23.  61 

15.  04 

23.  48 

15.  25 

23.  35 

15-  45 

23.  21 

15.66 

29 

24.  46 

15.58 

24.  32 

15.  97 

24.  18 

16-  01 

24.  04 

16.  22 

30 

25.  30 

16.  12 

25.  16 

16.  34 

25.  02 

16.56 

24.87 

16.78 

35 

29.  52 

18.  81 

29.  35 

19.  06 

29.  19 

19-  32 

29.  02 

19.57 

40 

33.74 

21.  49 

33.  55 

21.79 

33.  36 

22.  08 

33.  16 

22.  37 

45 

37.  95 

24.  18 

37.  74 

24.51 

37.52 

24.  84 

37.  31 

25.  16 

50 

42.  17 

26.  86 

41.  93 

27.  23 

41.  69 

27.  60 

41.  45 

27.  96 

55 

46.  39 

29.55 

46.  13 

29.  96 

45.  86 

30.  36 

45.  60 

30.  76 

60 

50.  60 

32.  24 

50.  32 

32.  68 

50.  08 

33.  12 

49.  74 

33.  55 

65 

54.  82 

34.  92 

54.  51 

35.  40 

54.  20 

35.  88 

53.  89 

36.  35 

70 

59.  04 

37.  61 

58.  72 

38.  12 

58.  37 

38.  64 

58.  03 

39.  14 

75 

63.  25 

40.  30 

62.  90 

40.  85 

62.  54 

41.  40 

62.  18 

41.  94 

80 

67.  47 

42.  98 

67.  09 

43.  57 

66.  71 

44.  15 

66.  32 

44.74 

85 

71.  69 

45.  67 

71  29 

46.  29 

70.  88 

46.  91 

70.  47 

47.  53 

90 

75.  91 

48.  36 

75.  48 

49.  02 

75.  05 

49.  67 

74.  61 

50.  33 

95 

80.  12 

51.04 

79.  67 

51.  74 

79.  22 

54.  43 

78.  76 

53.  12 

100 

84.  34 

53.  73 

83.  87 

54.  46 

83.  39 

55.  19 

82.  90 

55.  92 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

67%  Deg. 

67  Deg. 

66%  Deg. 

56  Deg. 

88          TRAVERSE  TABLE. 

b 

34^  Deg. 

36  Deg. 

35^  Deg. 

36  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  82 

0.  57 

0.  82 

0.  57 

0.  81 

0.  58 

0.  81 

0.  59 

2 

1.  65 

1.  13 

1.  64 

1.  15 

1.  63 

1.  16 

1.  62 

1.  18 

3 

2.  47 

1.  70 

2.  46 

1.  72 

2.  44 

1.  74 

2.  43 

1.  76 

4 

3.  30 

2.  27 

3.  28 

2.  29 

3.  26 

2.  32 

3.  24 

2.  35 

5 

4.  12 

2.  83 

4.  10 

2.  87 

4.  07 

2.  90 

4.  05 

2.  94 

6 

4.  94 

3.  40 

4.  91 

3.  44 

4.  88 

3.  48 

4.  85 

3.  53 

7 

5.77 

3.  96 

5.  73 

4.  01 

5.  70 

4.  06 

5.  66 

4.  11 

8 

6.  59 

4.  53 

6.  55 

4.  59 

6.  61 

4.  65 

6.  47 

4.  70 

9 

7.  42 

5.  10 

7.  37 

5.  16 

7.  33 

5.  23 

7.  28 

5.  29 

10 

8.24 

5.  66 

8.  19 

5.  74 

8.  14 

5.81 

8.  09 

5.  88 

11 

9.  07 

6.  23 

9.  01 

6.  31 

8.  96 

6.  39 

8.  90 

6.  47 

12 

9.89 

6.  80 

9.  83 

6.  88 

9.  77 

6.  97 

9.71 

7.  05 

13 

10.71 

7.  36 

10.  65 

7.  46 

10-  58 

7.55 

10.  52 

7.  64 

14 

11.  54 

7.  93 

11.  47 

8.  03 

11.  40 

8.  13 

11.  33 

8.  23 

15 

12.  36 

8.  50 

12.  29 

8.  60 

12.  21 

8.  71 

12.  14 

8.  82 

16 

13.  19 

9.  06 

13.  11 

9.  18 

13.  03 

9.  29 

12.  94 

9.  40 

17 

14.  01 

9.  63 

13.  93 

9.  75 

13.  84 

9.  87 

13.  75 

9.  99 

18 

14.  83 

10.  20 

14.  74 

10.  32 

14.  65 

10.  45 

14.  56 

10.  58 

19 

15.  66 

10.  76 

15.56 

10.  90 

15.  47 

11.  03 

15.  37 

11.  17 

20 

16.  48 

11.  33 

16.  38 

11.  47 

16.  28 

11.  61 

16.  18 

11.  76 

21 

17.  31 

11.  89 

17.  20 

12.  05 

17.  10 

12.  19 

16.  99 

12.  34 

22 

18.  13 

12.  46 

18.  02 

12.  62 

17.  91 

12.  78 

17.  80 

12.  93 

23 

18.  95 

13.  03 

18.  84 

13.  19 

18.  72 

13.  36 

18.  61 

13.  52 

24 

19.  78 

13.  59 

19.  66 

13.  77 

19.  54 

13.  94 

19.  42 

14.  11 

25 

20.  60 

14.  16 

20.  48 

14.  34 

20.  35 

14.  52 

20.  23 

14.  69 

26 

21.  43 

14.  73 

21.  30 

14.  91 

21.  17 

15.  10 

21.  03 

15.  28 

27 

22.  25 

15.  29 

22.  12 

15.  49 

21.  98 

15.  68 

21.  84 

15.  87 

28 

23.  08 

15.  86 

22.  94 

16.  06 

22.  80 

16.  26 

22.  65 

16.  46 

29 

23.  90 

16.  43 

23.  76 

16.  63 

23.  61 

16.  84 

23.  46 

17.  05 

•30 

24.  72 

16.  99 

24.  57 

17.21 

24.  42 

17.  42 

24.  27 

17.  63 

35 

28.  84 

19.  82 

28.  67 

20.  08 

28.  49 

20.  32 

28.  32 

20.57 

40 

32.  97 

22.  66 

32.  77 

22.  94 

32.  56 

23.  23 

32.  36 

23.  51 

45 

37.  09 

25.  49 

36.  86 

25.  81 

36.  64 

26.  13 

36.  41 

26.  45 

50 

41.21 

28.  32 

40.  96 

28.  68 

40.  71 

29.  04 

40.  45 

29.  39 

55 

45.  33 

31.  15 

45.  05 

31.55 

44.  78 

31.  94 

44.  50 

32.  23 

60 

49.  45 

33.  98 

49.  15 

34.  41 

48.  85 

34.  84 

48.54 

35.  27 

65 

53.  57 

36.  82 

53.  24 

37.28 

52.  92 

37.  75 

52.  59 

38.  21 

70 

57.  69 

39.  65 

57.  34 

40.  15 

56.  99 

40.  65 

56.  63 

41.  14 

75 

61.  81 

42.  48 

61.  44 

43.  02 

61.  06 

43.  55 

60.  68 

44.  08 

80 

65.  93 

45.  31 

65.  53 

45.  89 

65.  13 

46.  46 

64.  72 

47.  02 

85 

70.  05 

48.  14 

69.  63 

48.  75 

69.  20 

49.  36 

68.  77 

49.  96 

90 

74.  17 

50.  98 

73.  72 

51.62 

73.  27 

52.  26 

72.  81 

52.  90 

95 

78.  29 

53.81 

77.  82 

54.  49 

77.  34 

55.  17 

76.  86 

55.  84 

100 

82.  41 

56.  64 

81.  92 

57.  36 

81.  41 

58.  07 

80.  90 

58.  78 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

55V£  Deg. 

55  Deg. 

54K  Deg. 

54  Deg. 

TRAVERSE  TABLE.         89 

$ 

36J4  Deg. 

37  Deg. 

37%  Deg. 

38  Deg. 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  80 

0.  59 

0.  80 

0.  60 

0.  79 

0.  61 

0.  79 

0.  62 

2 

1.  61 

1.  19 

1.  60 

1.  20 

1.  59 

1.22 

1.  58 

1.  23 

3 

2.  41 

1.  78 

2.  40 

1.  81 

2.  38 

1.  83 

2.  36 

1.  85 

4 

3.  22 

2.  38 

3.  19 

2.  41 

3.  17 

2.  43 

3.  15 

2.  46 

5 

4.  02 

2.  97 

3.  99 

3.  01 

3.  97 

3-  04 

3.  94 

3.  08 

6 

4.  82 

3.  57 

4.  79 

3.  61 

4.  76 

3-  65 

4.  73 

3.  69 

7 

5.  63 

4.  16 

5.  59 

4.  21 

5.  55 

4-  20 

5.  52 

4.  31 

8 

6.  43 

4.  76 

6.  39 

4.  81 

6.  35 

4-  87 

6.  30 

4.  93 

9 

7.23 

5.  35 

7.  19 

5.  42 

7.  14 

5-  48 

7.  09 

5.  54 

10 

8.  04 

5.  95 

7.  99 

6.  02 

7.  93 

6.  09 

7.  88 

6.  16 

11 

8.  84 

6.  54 

8.  78 

6.  62 

8.  73 

6-  70 

8.  67 

6.  77 

12 

9.  65 

7.  14 

9.  58 

7.  22 

9.52 

7-  31 

9.  46 

7.  39 

13 

10.  45 

7.  73 

10.  38 

7.  82 

10.  31 

7-  91 

10.  24 

8.  00 

14 

11.  25 

8.  33 

11.  18 

8.  43 

11.  11 

8-52 

11.  03 

8.  62 

15 

12.  06 

8.  92 

11.  98 

9.  03 

11.  90 

9-  13 

11.  82 

9.23 

16 

12.  86 

9.  52 

12.  78 

9.  63 

12.  69 

9.  74 

12.  61 

9.  85 

17 

13.  67 

10.  11 

13.  58 

10.  23 

13.  49 

10-  35 

13.  40 

10.  47 

18 

14.  47 

10.  71 

14.  38 

10.  83 

14.  28 

10-  96 

14.  18 

11.  08 

19 

15.  27 

11.  30 

15.  17 

11.  43 

15.  07 

11-  57 

14.  97 

11.  70 

20 

16.  08 

11.  90 

15.  97 

12.  04 

15.  87 

12-  18 

15.  76 

12.  31 

21 

16.  88 

12.  49 

16.  77 

12.  64 

16.  66 

12-  78 

16.  55 

12.  93 

22 

17.  68 

13.  09 

17.  57 

13.  24 

17.  45 

13-  39 

17.  34 

13.  54 

23 

18.  49 

13.  68 

18.  37 

13.  84 

18.  25 

14-  00 

18.  12 

14.  16 

24 

19.  29 

14.  28 

19.  17 

14.  44 

19.  04 

14-  61 

18.  91 

14.  78 

25 

20.  10 

14.  87 

19.  97 

15.  05 

19.  83 

15-  22 

19.  70 

15.  39 

26 

20.  90 

15.  47 

20.  76 

15.  65 

20.  63 

15-  83 

20.  49 

16.  01 

27 

21.  70 

16.  06 

21.  56 

16.  25 

21.  42 

16-  44 

21.  28 

16.  62 

28 

22.  51 

16.  65 

22.  36 

16.  85 

22.  21 

17-  05 

22.  06 

17.  24 

29 

23.  31 

17.25 

23.  16 

17.  45 

23.  01 

17-  65 

22.  85 

17.  85 

30 

24.  12 

17.  84 

23.  96 

18.  05 

23.  80 

18-  26 

23.  64 

18.  47 

35 

28.  13 

20.  82 

27.  95 

21.  06 

27.  77 

21-  31 

27.  58 

21.  55 

40 

32.  15 

23.  79 

31.  95 

24.  07 

31.  73 

24-  35 

31.  52 

24.  63 

45 

36.  17 

26.  77 

35.  94 

27.  08 

35.  70 

27-  39 

35.  46 

27.  70 

50 

40.  19 

29.  74 

39.  93 

30.  09 

39.  67 

30-  44 

39.  40 

30.  78 

55 

44.  21 

32.  72 

43.  92 

33.  10 

43.  63 

33-  48 

43.  34 

33.  86 

60 

48.  23 

35.  69 

47.  92 

36.  11 

47.  60 

36-  53 

47.  28 

36.  94 

65 

52.  25 

38.  66 

51  91 

39.  121 

51.57 

39-  57 

51.  22 

40.  02 

70 

56.  27 

41.  64 

55.  90 

42.  13 

55.  53 

42.  61 

55.  16 

43.  10 

75 

60.  29 

44.  61 

59.  90 

45.  14 

59.50 

45.  66 

59.  10 

46.  17 

80 

64.  31 

47.  59 

63.  89 

48.  15 

63.  47 

48.  70 

63.  04 

49.  25 

85 

68.  33 

50.  56 

67.  48 

51.  15 

67.  43 

51.  74 

66.  98 

52.  33 

90 

72.  35 

53.  53 

71.  88 

54.  16 

71.  40 

54.  79  70.  92 

55.  41 

95 

76.  37 

56.  51 

75.  87 

57.  17 

75.  37 

57.  83  1 

74.  86 

58.  49 

100 

80.  39 

59.  48 

79.  86 

60.  18 

79.  34 

60.  88 

78.  80 

61.  57 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dop. 

Lat. 

53^  Deg. 

53  Deg. 

52^  Deg. 

62  Deg. 

90          TRAVERSE  TABLE. 

a 

38K  Deg. 

39  Deg. 

39^  Deg. 

40  Deg. 

1 

Lat.  |   Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  78 

0.  62 

0.  78 

0.  63 

0.  77 

0.  64 

0.  77 

0.  64 

2 

1.57 

1.  24 

1.  55 

1.  26 

1.  54 

1.27 

1.  53 

1.  29 

3 

2.  35 

1.  87 

2.  33 

1.  89 

2.  31 

1.  91 

2.  30 

1.  93 

4 

3.  13 

2.  49 

3.  11 

2.  52 

3.  09 

2.  54 

3.  06 

2.  57 

5 

3.  91 

3.  11 

3.  89 

3.  15 

3.  86 

3.  18 

3.  83 

3.  21 

6 

4.  70 

3.  74 

4.  66 

3.  98 

4.  63 

3.  82 

4.  60 

3.  86 

7 

5.  48 

4.  36 

5.  44 

4.  41 

5.  40 

4.  45 

5.  36 

4.  50 

8 

6.  26 

4.  98 

6.  22 

5.  03 

6.  17 

5.  09 

6.  13 

5.  14 

9 

7.  04 

5.  60 

6.99 

5.  66 

6.  94 

5.  72 

6.  89 

5.  79 

10 

7.  83 

6.  23 

7.  77 

6.  29 

7.  72 

6.  36 

7.  66 

6.  43 

11 

8.  61 

6.  85 

8.  55 

6.  92 

8.  49 

7.  00 

8.  43 

7.  07 

12 

9.  39 

7.  47 

9.  33 

7.  55 

9.  26 

7.  63 

9.  19 

7.  71 

13 

10.  17 

8.  09 

10.  10 

8.  18 

10.  03 

8.  27 

9.  96 

8.  36 

14 

10.  96 

8.  72 

10.  88 

8.  81 

10.  80 

8.  91 

10.  72 

9.  00 

15 

11.  74 

9.  34 

11.  66 

9.  44 

11.  57 

9.  54 

11.  49 

9.  64 

16 

12.  52 

9.  96 

12.  43 

10.  07 

12.  35 

10.  18 

12.  26 

10.  28 

17 

13.  30 

10.  58 

13.  21 

10.  70 

13.  12 

10.  81 

13.  02 

10.  93 

18 

14.  09 

11.  21 

13.  99 

11.  33 

13.  89 

11.  45 

13.  79 

11.  57 

19 

14.  87 

11.  83 

14.  77 

11.  96 

14.  66 

12.  09 

14.  55 

12.  21 

20 

15.  65 

12.  45 

15.  54 

12.  59 

15.  43 

12.  72 

15.  32 

12.  86 

21 

16.  43 

13.  07 

16.  32 

13.  22 

16.  20 

13.  36 

16.  09 

13.  50 

22 

17.  22 

13.  70 

17.  10 

13.  84 

16.  98 

13.  99 

16.  85 

14.  14 

23 

18.  00 

14.  32 

17.  87 

14.  47 

17.  75 

14.  63 

17.  62 

14.  78 

24 

18.  78 

14.  94 

18.  65 

15.  10 

18.  52 

15.  27 

18.  39 

15.  43 

25 

19.  57 

15.  56 

19.  43 

15.  73 

19.  29 

15.  90 

19.  15 

16.  07 

26 

20.  35 

16.  19 

20.  21 

16.  36 

20.  06 

16.  54 

19.  92 

16.  71 

27 

21.  13 

16.  81 

20.  98 

16.  99 

20.  83 

17.  17 

20.  68 

17.  36 

28 

21.  91 

17.  43 

21.  76 

17.  62 

21.  61 

17.  81 

21.  45 

18.  00 

29 

22.  70 

18.  05 

22.54 

18.  25 

22.  38 

18.  45 

22.  22 

18.  64 

30 

23.  48 

18.  68 

23.  31 

18.  88 

23.  15 

19.  08 

22.  98 

19.  28 

35 

27.  39 

21.  79 

27.  20 

22.  03 

27.  01 

22.  26 

26.  81 

22.  50 

40 

31.  30 

24.  90 

31.  09 

25.  17 

30.  86 

25.  44 

30.  64 

25.  71 

45 

35.  22 

28.  01 

34.  97 

28.  32 

34.  72 

28.  62 

34.  47 

28.  93 

50 

39.  13 

31.  13 

38.  86 

31.  47 

38.  58 

31.  80 

38.  30 

32.  14 

55 

43.  04 

34.  24 

42.  74 

34.  61 

42.  44 

34.  98 

42.  13 

35.  35 

60 

46.  96 

37.  35 

46.  63 

37.  76 

46.  30 

38.  16 

45.  96 

38.  57 

65 

50.  87 

40.  46 

50.51 

40.  91 

50.  16 

41.  35 

49.  79 

41.  78 

70 

54.  78 

43.  58 

54.  40 

44.  05 

54.  01 

44.  53 

53.  62 

45.  00 

75 

58.  70 

46.  69 

58.  29 

47.  20 

57.87 

47.  71 

57.  45 

48.  21 

80 

62.  61 

49.  80 

62.  17 

50.  35 

61.  73 

50.  89 

61.  28 

51.  42 

85 

66.  52 

52.  91 

66.  06 

53.  49 

65.  59 

54.  07 

65.  11 

54.  64 

90 

70.  43 

56.  03 

69.  94 

56.  64 

69.45 

57.  25 

68.  94 

57.  85 

95 

74.  35 

59.  14 

73.  83 

59.  79 

73.  30 

60.  43 

72.  77 

61.  06 

100 

78.  26 

62.  25 

77.  71 

62.  93 

77.  16 

63.  61 

76.  60 

64.  28 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

61  Y2  Deg. 

51  Deg. 

50%  Deg. 

50  Deg. 

TRAVERSE  TABLE.         91 

S 

40>£  Deg. 

41  Deg. 

41^  Deg. 

42  Deg. 

? 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.  76 

0.  65 

0.  75 

0  66 

0.  75 

0.  66 

0.  74 

0.  67 

2 

1.  52 

1.  30 

1.  51 

1.  31 

1.  50 

1.  33 

1.  49 

1.  34 

3 

2.  28 

1.  95 

2.  26 

1.  97 

2.25 

1.99 

2.  23 

2.  01 

4 

3.  04 

2.  60 

3.  02 

2.  62 

3.  00 

2.  65 

2.  97 

2.  68 

5 

3.  80 

3.  25 

3.77 

3.  28 

3.  74 

3-  31 

3.  72 

3.  35 

6 

4.  56 

3.  90 

4.  53 

3.  94 

4.  49 

3.  98 

4.  46 

4.  01 

7 

5.  32 

4.  55 

5.  28 

4.  59 

5.  24 

4-  64 

5.  20 

4.  68 

8 

6.  08 

5.20 

6.  04 

5.  25 

5.  99 

5.  30 

5.  95 

5.  35 

9 

6.  84 

5.  84 

6.79 

5.  90 

6.  74 

5.  96 

6.  69 

6.  02 

10 

7.  60 

6.  49 

7.  55 

6.  56 

7.  49 

6.  63 

7.  43 

6.  69 

11 

8.  36 

7.  14 

8.  30 

7.  22 

8.  24 

7.  29 

8.  17 

7.  36 

12 

9.  12 

7.  79 

9.  06 

7.  87 

8.  99 

7.  95 

8.  92 

8.  03 

13 

9.  89 

8.  44 

9.  81 

8,53 

9.  74 

8.  61 

9.  66 

8.  70 

14 

10.  65 

9.  09 

10.  57 

9.  18 

10.  49 

9.  28 

10.  40 

9.  37 

15 

11.  41 

9.  74 

11.  32 

9.  84 

11.  23 

9.  94 

11.  15 

10.  04 

16 

12.  17 

10.  39 

12.  08 

10.50 

11.  98 

10.  60 

11.  89 

10.  71 

17 

12.  93 

11.  04 

12.  83 

11.  15 

12.  73 

11-26 

12.  63 

11.  38 

18 

13.  69 

11.  69 

13.  58 

11.  81 

13.  48 

11-  93 

13.  38 

12.  04 

19 

14.  45 

12.  34 

14.  34 

12.  47 

14.  23 

12-59 

14.  12 

12.  71 

20 

15.  21 

12.  99 

15.  09 

13.  12 

14.  98 

13-  25 

14.  86 

13.  38 

21 

15.  97 

13.  64 

15.  85 

13.  78 

15.  73 

13-  91 

15.  61 

14.  05 

22 

16.  73 

14.  29 

16.  60 

14.  43 

16.  48 

14-  58 

16.  35 

14.  72 

23 

17.  49 

14.  94 

17.  36 

15.  09 

17.  23 

15-24 

17.  09 

15.  39 

24 

18.  25 

15.  59 

18.  11 

15.  75 

17.  97 

15.  90 

17.  84 

16.  06 

25 

19.  01 

16.  24 

18.  87 

16.  40 

18.  72 

16-  57 

18.  58 

16.  73 

26 

19.  77 

16.  89 

19.  62 

17.  06 

19.  47 

17.  23 

19.  32 

17.  40 

27 

20.  53 

17.  54 

20.  38 

17.  71 

20.  22 

17-  89 

20.  06 

18.  07 

28 

21.  29 

18.  18 

21.  13 

18.  37 

20.97 

18-  55 

20.  81 

18.  74 

29 

22.  05 

18.  83 

21.  89 

19.  03 

21.72 

19.  22 

21.  55 

19.  40 

30 

22.  81 

19.  48 

22.  64 

19.  68 

22.  47 

19-  88 

22.  29 

20.  07 

35 

26.  61 

22.  73 

26.  41 

22.  96 

26.  21 

23-  19  26.  01 

23.  42 

40 

30.  42 

25.  98 

30.  19 

26.  24 

29.  96 

26-  50  29.  73  26.  77 

45 

34.  22 

29.  23 

33.  96 

29.  52 

33.  70 

29-  82 

33.  44 

30.  11 

50 

38.  02 

32.  47 

37.  74 

32.  80 

37.  45 

33.  13 

37.  16 

33.  46 

55 

41.  82 

35.  72 

41.  51 

36.  08 

41.  19 

36-  44 

40.  87 

36.  80 

60 

45.  62 

38.  97 

45.  28 

39.  36 

44.  94 

39.  76 

44.  59 

40.  15 

65 

49.  43 

42.  21 

49.  06 

42.  64 

48.  68 

43-  07 

48.  30 

43.  49 

70 

53.  23 

45.  46 

52.  83 

45.  92 

52.  43 

46.  38 

52.  02 

46.  84 

75 

57.  03 

48.  71 

56.  60 

49.  20 

56.  17 

49.  70 

55.  74 

50.  18 

80 

60.  83 

51.  96 

60.  38 

52.  48 

59.  92 

53.  01 

59.  45 

53.  53 

85 

64.  63 

55.  20 

64.  15 

55.  76 

63.  66 

56.  32 

63.  17 

56.  88 

90 

68.  44 

58.  45 

67.  92 

59.  05 

67.  41 

59.  64  66.  88 

60.  22 

95 

72.  24 

61.  70 

71.  70 

62.  33 

71.  15 

62.  95 

70.  60 

63.  57 

100 

76.  04 

64.  98 

75.  47 

65.  61 

74.  90 

66.  26 

74.  31 

66.  91 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Top. 

Lat 

49^  Deg. 

49  Deg. 

48^  Deg. 

48  Deg. 

22 


92          TRAVERSE  TABLE. 

a 

42%  Deg. 

43  Deg. 

43^  Dog. 

44  Deg. 

i 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

0.  74 

0.  68 

0.  73 

0.  68 

0.  73 

0.  69 

0.  72 

0.  69 

2 

1.47 

1.  35 

1.  46 

1.  36 

1.  45 

1.  38 

1.  44 

1.  39 

3 

2.  21 

2.  03 

2.  19 

2.  05 

2.  18 

2.07 

2.  16 

2.  08 

4 

2.  95 

2.  70 

2.  93 

2.  73 

2.  90 

2.  75 

2.  88 

2.  78 

5 

3.  69 

3.  38 

3.  66 

3.  41 

3.  63 

3.44 

3.  60 

3.  47 

6 

4.  42 

4.  05 

4.  39 

4.  09 

4.  35 

4.  13 

4.  32 

4.  17 

7 

5.  16 

4.  73 

5.  12 

4.  77 

5.  08 

4.  82 

5.  04 

4.  86 

8 

5.  90 

5.  40 

5.  85 

5.  46 

5.  80 

5.51 

5.  75 

5.  56 

9 

6.64 

6.  08 

6.  58 

6.  14 

6.  53 

6.  20 

6.  47 

6.  25 

10 

7.  37 

6.  76 

7.  31 

6.82 

7.  25 

6.  88 

7.  19 

6.  95 

11 

8.  11 

7.  43 

8.  04 

7.  50 

7.  98 

7.  57 

7.  91 

7.  64 

12 

8.  85 

8.  11 

8.  78 

8.  18 

8.  70 

8.  26 

8.  63 

8.  34 

13 

9.58 

8.  78 

9.51 

8.  87 

9.  43 

8.  95 

9.  35 

9.  03 

14 

10.  32 

9.  46 

10.  24 

9.  55 

10.  16 

9.  64 

10.  07 

9.  73 

15 

11.  06 

10.  13 

10.  97 

10.  23 

10.  88 

10.  33 

10.  79 

10.  42 

16 

11.  80 

10.  81 

11.  70 

10.  91 

11.  61 

11.  01 

11.  51 

11.  11 

17 

12.  53 

11.  48 

12.  43 

11.  59 

12.  33 

11.  70 

12.  23 

11.  81 

18 

13.  27 

12.  16 

13.  16 

12.  28 

13.  06 

12.  39 

12.  95 

12.  50 

19 

14.  01 

12.  84 

13.  90 

12.  96 

13.  78 

13.  08 

13.  67 

13.  20 

20 

14.  75 

13.  51 

14.  63 

13.  64 

14.  51 

13.  77 

14.  39 

13.  89 

21 

15.  48 

14.  19 

15.  36 

14.  32 

15.  23 

14.  46 

15.  11 

14.  59 

22 

16.  22 

14.  86 

16.  09 

15.  00 

15.  96 

15.  14 

15.83 

15.28 

23 

16.  96 

15.  54 

16.  82 

15.  69 

16.  88 

15.  83 

16.  54 

15.  98 

24 

17.  69 

16.  21 

17.  55 

16.  37 

17.  41 

16.  52 

17.26 

16.  67 

25 

18.  43 

16.  89 

18.  28 

17.  05 

18.  13 

17.21 

17.  98 

17.  37 

26 

19.  17 

17.  57 

19.  02 

17.  73 

18.  86 

17.  90 

18.  70 

18.  06 

27 

19.  91 

18.  24 

19.  75 

18.  41 

19.  59 

18.  59 

19.  42 

18.  76 

28 

20.  64 

18.  92 

20.  48 

19.  10 

20.  31 

19.  27 

20.  14 

19.  45 

29 

21.  38 

19.  59 

21.  21 

19.  78 

21.  04 

19.  96 

20.  86 

20.  15 

30 

22.  12 

20.  27 

21.  94 

20.  46 

21.76 

20.  65 

21.58 

20.  84 

35 

25.  80 

23.  65 

25.  60 

23.  87 

25.  39 

24.  09 

25.  18 

24.  31 

40 

29.  49 

27.  02 

29.  25 

27.  28 

29.  01 

27.  53 

28.  77 

27.  79 

45 

33.  18 

30.  40 

32.  91 

30.  69 

32.  64 

30.  98 

32.  37 

31.26 

50 

36.  86 

33.  78 

36.  57 

34.  10 

36.27 

34.  42 

35.  57 

34.  73 

55 

40.  55 

37.  16 

40.  22 

37.  51 

39.  90 

37.  86 

39.  96 

38.  21 

60 

44.  24 

40.  54 

43.  88 

40.  92 

43.52 

41.  30 

43.  16 

41.68 

65 

47.  92 

43.  91 

47.  54 

44.  33 

47.  15 

44.  74 

46.  76 

45.  15 

70 

51.  61 

47.  29 

51.  19 

47.  74 

50.78 

48.  18 

50.  35 

48.  63 

75 

55.  30 

50.  67 

54.  85 

51.  15 

54.  40 

51.  63 

53.  95 

52.  10 

80 

58.  98 

54.  05 

58.  51 

54.  56 

58.  03 

55.  07 

57.  55 

55.  57 

85 

62.  67 

57.  43 

62.  17 

57.  97 

61.66 

58.  51 

61.  14 

59.  05 

90 

66.  35 

60.  80 

65.82 

61.  38 

65.28 

61.  95 

64.  74 

62.52 

95 

70.  04 

64.  18 

69.  48 

64.  79 

68.  91 

65.  39 

68.  34 

65.  99 

100 

73.  73 

67.  56 

73.  14 

68.  20 

72.  54 

68.  84 

71.  93 

69.  47 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

47%  Deg. 

47  Deg. 

46%  Deg. 

46  Deg. 

TRAVERSE  TABLE.         93 

2 

44^  Deg. 

45  Deg. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.71 

0.  70 

0.  71 

0  71 

2 

1.43 

1.  40 

1.  41 

1.  41 

3 

2.  14 

2.  10 

2.  12 

2.  12 

4 

2.  85 

2.  80 

2.  83 

2.  83 

5 

3.  57 

3.  50 

3.  54 

3.  54 

6 

4.  28 

4.  21 

4.  24 

4.  24 

7 

4.  99 

4.  91 

4.  95 

4.  95 

8 

5.  71 

5.  61 

5.  66 

5.  66 

, 

9 

6.  42 

6.  31 

6.  36 

6.  36 

10 

7.  13 

7.  01 

7.  07 

7.  07 

11 

7.  85 

7.  71 

7.  78 

7.  78 

12 

8.  56 

8.  41 

8.  49 

8.  49 

13 

9.  27 

9.  11 

9.  19 

9.  19 

14 

9.  99 

9.  81 

9.  90 

9.  90 

15 

10.  70 

10.  51 

10.  61 

10.  61 

16 

11.  41 

11.  21 

11.  31 

11.  31 

17 

12.  13 

11.  92 

12.  02 

12.  02 

18 

12.  84 

12.  62 

12.  73 

12.  73 

19 

13.  55 

13.  32 

13.  43 

13.  43 

20 

14.  26 

14.  02 

14.  14 

14.  14 

21 

14.  98 

14.  72 

14.  85 

14.  85 

22 

15.  69 

15.  42 

15.  56 

15.  56 

23 

16.  40 

16.  12 

16.  26 

16.  26 

24 

17.  12 

16.  82 

16.  97 

16.  97 

25 

17.  83 

17.  52 

17.  68 

17.  68 

26 

18.54 

18.  22 

18.  38 

18.  38 

27 

19.26 

18.  92 

19.  09 

19.  09 

28 

19.  97 

19.  63 

19.  80 

19.  80 

29 

20.  68 

20.  33 

20.  51 

20.  51 

30 

21.  40 

21.  03 

21.  21 

21.21 

35 

24.  96 

24.  53 

24.  75 

24.  75 

40 

28.  53 

28.  04 

28.  28 

28.  28 

45 

32.  10 

31.  54 

31.  82 

31.  82 

50 

35.  66 

35.  05 

35.  36 

35.  36 

55 

39.  23 

38.  55 

38.  89 

38.  89 

60 

42.  79 

42.  05 

42.  43 

42.  43 

65 

46.  36 

45.  56 

45.  96 

45.  96 

70 

49.  93 

49.  06 

49.  50 

49.  50 

75 

53.  49 

52.  57 

53.  03 

53.  03 

80 

57.  06 

56.  07 

56.  57 

56.  57 

85 

60.  63 

59.  58 

60.  10 

60.  10 

90 

64.  19 

63.  08 

63.  64 

63.  64 

95 

67.  76 

66.  59 

67.  18 

67.  18 

100 

71.  33 

70.  09 

70.  71 

70.  71 

Dep. 

Lat. 

Dep. 

Lat. 

45^  Deg. 

45  Deg. 

94                     Meiidianal  Parts.                TABLE  IV. 

/ 

0° 

1 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

14° 

15° 

0 

0 

60 

120 

180 

240 

SOU 

361 

421 

481 

542 

603 

664 

725 

787 

8T8 

910 

1 

1 

61 

121 

181 

241 

301 

362 

422 

483 

543 

604 

665 

726 

788 

850 

911 

2 

L1 

62 

122 

182 

242 

302 

363 

423 

484 

544 

605 

666 

727 

789 

851 

913 

3 

3 

63 

123 

183 

243 

303 

364 

424 

485 

545 

606 

667 

728 

790 

852 

914 

4 

4 

64 

124 

184 

244 

304 

365 

425 

486 

546 

6-07 

668 

729 

791 

853 

915 

6 

6 

65 

125 

185 

245 

305 

366 

426 

487 

547 

608 

669 

730 

792 

854 

916 

6 

(i 

66 

128 

186 

246 

303 

367 

427 

488 

548 

609 

670 

731 

793 

855 

917 

7 

7 

6? 

127 

187 

247 

307 

368 

428 

489 

549 

610 

671 

732 

794 

856 

918 

8 

8 

68 

128 

188 

248 

308 

369 

429 

490 

550 

611 

672 

734 

795 

857 

919 

9 

g 

69 

129 

189 

249 

309 

370 

430 

491 

551 

612 

673 

735 

796 

858 

920 

10 

10 

70 

130 

190 

250 

310 

371 

431 

492 

552 

613 

674 

736 

797 

859 

921 

11 

n 

71 

131 

191 

251 

311 

372 

432 

493 

553 

614 

675 

737 

798 

860 

922 

12 

L2 

72 

132 

192 

252 

312 

373 

433 

494 

554 

615 

676 

738 

799 

861 

923 

13 

13 

73 

133 

193 

253 

313 

374 

434 

495 

555 

616 

677 

739 

800 

862 

924 

14 

14 

74 

134 

194 

254 

314 

375 

435 

496 

556 

617 

678 

740 

801 

863 

925 

15 

15 

75 

135 

195 

255 

315 

376 

436 

497 

557 

618 

679 

741 

802 

864 

926 

16 

16 

76 

136 

196 

256 

316 

377 

437 

468 

558 

619 

680 

742 

803 

865 

927 

17 

17 

77 

137 

197 

257 

317 

378 

438 

499 

559 

620 

681 

743 

804 

366 

928 

18 

L8 

78 

133 

198 

258 

318 

379 

439 

500 

5SO 

621 

682 

744 

805 

867 

929 

19 

19 

79 

139 

199 

259 

319 

380 

440 

501 

561 

622 

683 

745 

806 

868 

930 

20 

20 

80 

140 

200 

260 

320 

381 

441 

502 

562 

623 

684 

746 

807 

869 

931 

21 

•3! 

81 

141 

201 

261 

321 

382 

442 

503 

564 

624 

685 

747 

808 

870 

932 

22 

22 

82 

142 

202 

262 

322 

383 

443 

504 

565 

625 

687 

748 

809 

871 

933 

23 

23 

83 

143 

203 

263 

323 

384 

444 

505 

566 

626 

688 

749 

810 

872 

934 

24 

24 

84 

144 

204 

264 

824 

385 

445 

506 

567 

627 

689 

750 

811 

873 

935 

25 

26 

85 

145 

205 

265 

325 

386 

446 

507 

568 

628 

690 

751 

812 

874 

936 

26 

26 

86 

146 

206 

266 

326 

387 

447 

508 

569 

629 

691 

752 

313 

875 

937 

27 

27 

87 

147 

20? 

267 

327 

388 

448 

509 

570 

631 

692 

753 

815 

876 

938 

28 

28 

88 

148 

208 

268 

328 

389 

449 

510 

571 

632 

693 

754 

816 

877 

939 

29 

29 

89 

149 

209 

269 

330 

390 

450 

511 

572 

633 

694 

755 

817 

878 

941 

30 

30 

90 

150 

210 

270 

331 

391 

451 

512 

573 

634 

695 

756 

818 

879 

942 

31 

ol 

91 

151 

211 

271 

332 

392 

452 

613 

574 

655 

698 

757 

819 

880 

943 

32 

32 

92 

152 

212 

272 

333 

393 

453 

514 

575 

636 

697 

758 

820 

882 

944 

33 

33 

93 

153 

213 

273 

334 

394 

454 

515 

576 

637 

698 

759 

821 

883 

945 

34 

B4 

94 

154 

214 

274 

335 

395 

455 

516 

577 

638 

699 

760 

822 

884 

946 

35 

36 

95 

155 

215 

275 

336 

396 

456 

517 

578 

639 

700 

761 

823 

885 

947 

36 

36 

96 

15G 

216 

276 

337 

397 

457 

518 

579 

640 

701 

762 

824 

886 

948 

37 

37 

97 

157 

217 

277 

338 

398 

458 

519 

580 

941 

702 

763 

825 

887 

949 

38 

38 

98 

158 

218 

278 

339 

399 

459 

520 

581 

642 

703 

764 

826 

888 

950 

39 

39 

99 

159 

219 

279 

340 

400 

460 

521 

582 

643 

704 

765 

827 

889 

951 

40 

40 

100 

160 

220 

280 

341 

401 

461 

522 

583 

644 

705 

766 

828 

890 

952 

41 

•11 

101 

161 

221 

281 

342 

402 

462 

523 

584 

645 

706 

767 

829 

891 

953 

42 

42 

102 

162 

22-2 

282 

343 

403 

463 

524 

585 

646 

707 

768 

830 

892 

954 

43 

43 

103 

163 

223 

283 

344 

404 

4641  525 

586 

647 

708 

769 

831 

893 

955 

44 

44 

104 

164 

224 

284 

345 

405 

465 

523 

587 

b'48 

709 

770 

832 

894 

956 

45 

!.') 

105 

165 

225 

285 

346 

406 

466 

527 

588 

649 

710 

771  833 

895 

957 

46 

46 

106 

166 

226 

286 

347 

407 

467 

528 

589 

650 

711 

772 

834 

896 

958 

47 

47 

107 

167 

227 

287 

348 

408 

468 

529 

590 

651 

712 

773 

835 

897 

959 

48 

48 

108 

16S 

228 

288 

349 

409 

469 

530 

591 

652 

713 

774 

836 

898 

9GO 

49 

49 

109 

169 

229 

289 

350 

410 

470 

631 

592 

653 

714 

775 

837 

899 

961 

50 

50 

110 

170 

230 

290 

351 

411 

471 

532 

593 

654 

715 

777 

838 

900 

962 

51 

51 

111 

171 

231 

291 

352 

412 

472 

533 

594 

655 

716 

778 

839 

901 

963 

52 

52 

112 

172 

232 

292 

353 

413 

473 

534 

595 

656 

717 

779 

840 

902 

964 

53 

:,;: 

113 

173 

233 

293 

354 

414 

474 

535 

596 

657 

718 

780 

841 

903 

965 

54 

54 

114 

174 

234 

294 

355 

415 

476 

536 

597 

668 

719 

781 

842 

904 

966 

55 

55 

115 

175 

235 

295 

356 

416 

477 

53  7 

598 

659 

720 

782 

843 

905 

968 

56 

56 

116 

176 

236 

296 

357 

417 

478 

538 

599 

660 

721 

783 

844 

606 

969 

57 

57 

117 

177 

237 

297 

358 

418 

479 

539 

600 

661 

722 

784 

845 

907 

970 

58 

58 

118 

178 

288 

298 

359 

419 

480 

540 

601 

662 

723 

785 

846 

908 

971 

59 

59 

119 

179 

239 

299 

360 

420 

481 

541 

602 

663 

754 

786 

847 

909 

972 

TABLE  IV.               Meridianal  Parts.                      95 

0 

10° 

"973 

17° 
1035 

18.J 
1098 

19° 

1161 

20° 
1225 

21° 

1289 

22° 
1354 

233 
14i9 

24° 

1484 

to0 

1550 

2tJ° 
16T6 

27° 
1684 

28° 
1751 

1 

974 

1036 

1099 

1163 

1226 

1290 

1355 

1420 

1485 

1551 

1618 

1685 

1752 

2 

975 

1037 

1100 

1164 

1227 

1291 

1356 

1421 

1486 

1552 

1619 

1686 

1753 

3 

976 

1038 

1101 

1165 

1228 

1292 

1357 

1422 

1487 

1553 

1620 

1687 

1755 

4 

977 

1039 

1102 

1166 

1229 

1293 

1358 

1423 

1488 

1554 

1621 

1688 

1756 

6 

978 

1041 

1103 

1167 

1230 

1295 

1359 

1424 

1490 

1556 

1622 

1689 

1757 

6 

979 

1042 

1105 

116S 

1232 

1296 

1360 

1425 

1491 

1557 

1623 

1690 

1758 

7 

980 

1043 

1108 

1169 

1233 

1297 

1361 

1426 

1492 

1558 

1624 

1692 

1759 

8 

981 

1044 

1107 

1170 

1234 

1298 

1362 

1427 

1493 

1559 

1625 

1693 

1760 

9 

982 

1045 

1108 

1171 

1235 

1299 

1363 

1428 

1494 

1560 

1626 

1694 

1761 

10 

983 

1046 

1109 

1172 

1236 

1300 

1364 

1430 

1495 

1561 

1628 

1695 

1762 

11 

984 

1047 

1110 

1173 

1237 

1301 

1366 

1431 

1496 

1562 

1629 

1696 

1764 

12 

985 

1048 

1111 

1174 

1238 

1302 

1367 

1432 

1497 

1563 

1630 

1697 

1765 

13 

986 

1049 

1112 

1175 

1239 

1303 

1368 

1433!  1498 

1564 

1631 

1698 

1766 

14 

987 

1050 

1113 

1176 

1240 

1304 

1369 

1434J  1499 

1565 

1632 

1699 

1767 

15 

988 

1051 

1114 

1177 

1241 

1305 

1370 

14351  1500 

1567 

1633 

1700 

1768 

16 

989 

1052 

1115 

1178 

1242 

1306 

1371 

1436 

1502 

1568 

1634 

1701 

1769 

17 

990 

1053 

1116 

1179 

1243 

1307 

1372 

1437 

1503 

1569 

1635 

1703 

1770 

18 

991 

1054 

1117 

1181 

1244 

1308 

1373 

1438 

1504 

1570 

1637 

1704 

1772 

19 

993 

1055 

1118 

1182 

1245 

1310 

1374 

1439 

1505 

1571 

1638 

1705 

1773 

20 

994 

1056 

1119 

1183 

1246 

1311 

1375 

1440 

1506 

1572 

1639 

1706 

1774 

21 

995 

1057 

1120 

1184 

1248 

1312 

1376 

1441  i  1507 

1673 

1640 

1707 

1775 

22 

996 

1058 

1121 

1185 

1249 

1313 

1377 

14431  1508 

1574 

1641 

1708 

1776 

^:? 

997 

1059 

1122 

1186 

1250 

1314 

1379 

1444 

1509 

1575 

1642 

1709 

1777 

24 

99S 

1060 

1123 

1187 

1251 

1315 

1380 

1445 

1510 

1577 

1643 

1711 

1778 

25 

999 

UG1 

1125 

1188 

1252 

1316 

1381 

1446 

1511 

1578 

1644 

1712 

1780 

26 

1000 

1063 

1126 

1189 

1253 

1317 

1382 

1447 

1513 

1579 

1645 

1713 

1781 

27 

1001 

1064 

1127 

1190 

1254 

1318 

1383 

1448 

1514 

1580  1647 

1714 

1782 

28 

1002 

1065 

1128 

1191 

1255 

1319 

1384 

1449 

1515 

1581  1648 

1715 

1783 

29 

1003 

1066 

1129 

1192 

1256 

1320 

1385 

1450 

1516 

1582  1649 

1716 

1784 

30 

1004 

1067 

1130 

1193  1257 

1321 

1386 

1451 

1517 

1583  1650 

1717 

1785 

31 

1005 

1068 

1131 

1194  1258 

1322 

1387 

1452 

1518 

1584 

1661!  1718 

1786 

32 

1006 

1069 

1132 

1195  1259 

1324 

1388 

1453 

1519 

1585 

1652 

1720 

1787 

33 

1007 

1070 

1133 

1196 

1260 

1325 

1389 

1455 

1520 

1586 

1653 

1721 

1789 

34 

1008 

1071 

1134 

1198 

1261 

1326 

1390 

1456 

1521 

1588 

1654 

1722 

1790 

35 

1009 

1072 

1135 

1199|  1262 

1327 

1392 

1457 

1522 

1589 

1656 

1723 

1791 

36 

1010 

1073 

1136 

1200i  1264 

1328 

1393 

1458 

1524 

1690 

1657 

1724 

1792  1 

3? 

1011 

1074 

1137 

1201!  1265 

1329 

1394 

1459 

1525 

1691 

1658 

1725 

1793 

38 

1012 

1075 

1138 

1202  1266 

1330 

1395 

1460 

1526 

1592 

1659 

1726 

1794 

39 

1013 

1076 

1139 

1203 

1267 

1331 

1396 

1461 

1527 

1593 

1660 

1727 

1795 

40 

1014 

1077 

1140 

1204 

1268 

1332 

1397 

1462 

1528 

1694 

1661 

1729 

1797 

41 

1015 

1078 

1141 

1205  1269 

1333 

1398 

1463 

1529 

1595 

1662 

1730 

1798 

42 

1016 

1079 

1142 

l-'Oo  1-270 

1334 

1399 

1464 

1530 

1596 

1663 

1731 

1799 

43 

1018 

1080 

1144 

120? 

1271 

1335 

1400 

1465 

1531 

1598 

1664 

1732 

1800 

44 

1019 

1081 

1145 

1208 

1272 

1336 

1401 

1467 

1532 

1599  1666 

1733 

1801 

45 

1020 

1082 

1146 

1209|  1273 

1338 

1402 

1468 

1533 

1600  1667 

1734 

1802 

46 

1021 

1084 

1147 

1210  1274 

1339 

1403 

1469 

1535 

1601 

1668 

1735 

1803 

47 

102-2 

1085 

1148 

1211  1275 

1340 

1405 

1470 

1536 

1602 

1669 

1736 

1805 

48 

1023 

1086 

1149 

1212  1276 

1341 

1406 

1471 

1  ,37 

1603 

1670 

1737 

1806 

49 

1024 

1087 

1150 

0213  1277 

1342 

1407 

1472 

1538 

1604 

1671 

1739 

1807 

50 

1025 

1038 

1151 

1215  1278 

1343 

1408 

1473 

1539 

1605 

1672 

1740 

1808 

51 

1026 

1089 

1152 

1216  1-280 

1344 

1409 

1474 

1540 

1606 

1673 

1741 

1809 

52 

1027 

1090 

1153 

1217  1281 

1345 

1410 

1476 

1541 

160S 

1675 

1742 

1810 

53 

1028 

1091 

1154 

1218 

1282 

1346 

141  ll  1476 

1542 

1609 

1676 

1743 

1811 

54 

1029 

1092 

1155 

1219 

1283 

1347 

1412 

1477 

1543 

1610 

1677 

1744 

1813 

55 

1030 

1093 

1156 

1220 

1284 

1348 

1413 

1479 

1544 

1611 

1678 

1746 

1814 

56 

1031 

1094 

1157 

1221 

1285 

1349 

1414 

1480 

1546 

1612 

1679 

1747 

1815 

57 

1032 

1095 

1158 

1222 

1286 

1350 

1415 

1481 

1647 

1613 

1680 

1748 

1816 

58 

1033 

1096 

1159 

1223 

1287 

1352 

1416 

1482 

1548 

1614 

1681 

1749 

1817 

59 

1034 

1097 

1160 

1224 

1288 

1353 

1418 

1483 

1549 

1615 

1682 

1760 

1818 

96                      Meridian*!  Parts,               TABLH  IV. 

' 

29° 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

391 

40° 

41° 

~o 

1819 

1888 

1958 

2028 

2~100 

2171 

2244 

23T8 

2393 

2ley 

2545 

2623 

2702 

1 

1821 

1890 

1959 

2030 

2101 

2173 

2246 

2319 

2394 

2470 

2546 

2624 

2703 

2 

1822 

1891 

1960 

2031 

2102 

2174 

2247 

2320 

2395 

2471 

2648 

2625 

2704 

3 

1823 

1892 

1962 

2032 

2103 

2175 

2248 

2322 

2396 

2472 

2549 

2627 

2706 

4 

1924 

1893 

1963 

2033 

2104 

2176 

2249 

2323 

2398 

2473 

2550 

2628 

2707 

6 

1825 

1894 

1964 

2034 

2105 

2178 

2250 

2324 

2399 

2475 

2551 

2629 

2708 

6 

1826 

1895 

1965 

2035 

2107 

2179 

2252 

2325 

2400 

2479 

2553 

2631 

2710 

7 

1827 

1896 

1966 

2037 

2108 

2180 

2253 

2327 

2401 

2477 

2554 

2632 

2711 

8 

1829 

1898 

1967 

2038 

2109 

2181 

2254 

2328 

2403 

2478 

2555 

2633 

2712 

9 

1830 

1899 

1969 

2039 

2110 

2182 

2255 

2329 

2404 

2480 

2557 

2634 

2714 

10 

1831 

1900 

1970 

2040 

2111 

2184 

2257 

2330 

2405 

2481 

2558 

2636 

2715 

11 

1832 

1901 

1971 

2041 

2113 

2185 

2258 

2332 

2406 

2482 

2559 

2637 

2716 

12 

1833 

1902 

1972 

2043 

2114 

2186 

2259 

2333 

2408 

2484 

2560 

2638 

2718 

13 

1834 

1903 

1973 

2044 

2115 

2187 

2260 

2334 

2409 

2485 

2562 

2640 

2719 

14 

1835 

1905 

1974 

2045 

2116 

2188 

2261 

2335 

2410 

2486 

2563 

2641 

2720 

15 

1837 

1906 

1976 

2046 

2117 

2190 

2263 

2337 

2411 

2487 

2564 

2642 

2722 

16 

1838 

1907 

1977 

2047 

2119 

2191 

2264 

2338 

2413 

2489 

2566 

2644 

2723 

17 

1839 

1908 

1978 

2048 

2120 

2192 

2265 

2339 

2414 

2490 

2567 

2645 

2724 

18 

1840 

1909 

1679 

2050 

2121 

2193 

2266 

2340 

2416 

2491 

2568 

2646 

2726 

19 

1841 

1910 

1980 

2051 

2122 

2194 

2268 

2342 

2416 

2492 

2569 

2648 

2727 

20 

1842 

1912 

1981 

2052 

2123 

2196 

2269 

2343 

2418 

2494 

2571 

2649 

2728 

21 

1843 

1913 

1983 

2053 

2125 

2197 

2270 

2344 

2419 

2495 

2572 

2650 

2729 

22 

1845 

1914 

1984 

2054 

2126 

2198 

2271 

2345 

2420 

2496 

2573 

2551 

2731 

23 

1846 

1915 

1985 

2056 

2127 

2199 

2272 

2346 

2422 

2498 

2575 

2653 

2732 

24 

1847 

1916 

1986 

2057 

2128 

2200 

2274 

2348 

2423 

2499 

2576 

2654 

2733 

26 

1848 

1917 

1987 

2058 

2129 

2202 

2275 

2349 

2424 

2500 

2577 

2655 

2735 

29 

1849 

1918 

1988 

2059 

2131 

2203 

2266 

2350 

2425 

2501 

2578 

2657 

2736 

27 

1850 

1920 

1990 

2060 

2132 

2204 

2267 

2351 

2427 

2503 

2580 

2658 

2737 

28 

1852 

1921 

1991 

2061 

2133 

2205 

2279 

2353 

2428 

2504 

2581 

2659 

2739 

29 

1853 

1922 

1992 

2063 

2134 

2207 

2280 

2354 

2429 

2505 

2582 

2661 

2740 

30 

1854 

1923 

1993 

2064 

2135 

2208 

2281 

2355 

2430 

2506 

2584 

26S2 

2742 

31 

1855 

1924 

1994 

2065 

2137 

2209 

2232 

2356 

2432 

2508 

25S5 

2663 

2743 

32 

1856 

1925 

1995 

2066 

2138 

2210 

2283 

2358 

2433 

2509 

2586 

2665 

2744 

33 

1857 

1927 

1997 

2067 

2139 

2211 

2285 

2359 

2434 

2510 

2588 

2666 

2746 

34 

1858 

1928 

1998 

2069 

2140 

2213 

2286 

2360 

2435 

2512 

2589 

2667 

2747 

35 

1360 

1929 

1999 

2070 

2141 

2214 

2287 

2361 

2437 

2513 

2590 

2669 

2748 

36 

1861 

1930 

2000 

2071 

2143 

2215 

2288 

2363 

2438 

2514 

2591 

2670 

2750 

37 

1892 

1931 

2001 

2072 

2144 

2216 

2290 

2364 

2439 

2515 

2593 

2671 

2751 

38 

1863 

1932 

2002 

2073 

2146 

2217 

2291 

2365 

2440 

2517 

2594 

2673 

2752 

39 

1864 

1934 

2004 

2075 

2146 

2219 

2292 

2366 

2442 

2518 

2595 

2674 

2754 

40 

1865 

1935 

2005 

2076 

2147 

2220 

2293 

2368 

2443 

2519 

2597 

2675 

2755 

41 

1866 

1936 

2006 

2077 

2149 

2221 

2295 

2369 

2444 

2521 

2598 

2676 

2756 

42 

1808 

1937 

2077 

2078 

2150 

2222 

2296 

2370 

2445 

2522 

2599 

2678 

2758 

43 

1869 

1938 

2008 

2079 

2151 

2224 

2297 

2371 

2447 

2523 

2601 

2679 

2759 

44 

1870 

1939 

2010 

2080 

2152 

2225 

2298 

2373 

2448 

2524 

2602 

2680 

2760 

45 

1871 

1941 

2011 

2082 

2153 

2226 

2299 

2374 

2449 

2526 

2603 

2682 

2762 

46 

1872 

1942 

2012 

2083 

2155 

2227 

2301 

2375 

2451 

2527 

2604 

2683 

2763 

47 

1873 

1943 

2013 

2084 

2156 

2228 

2302 

2376 

2452 

2528 

2606 

2684 

2764 

48 

1876 

1944 

2014 

2085 

2157 

2230 

2303 

2378 

2453 

2530 

2907 

2636 

2766 

49 

1876 

1945 

2015 

2086 

2J5S 

2231 

2304 

2379 

2454 

2531 

2608 

2687 

2767 

50 

1877 

1946 

2017 

2088 

2159 

2232 

2306 

2380 

2456 

2532 

2610 

2688 

2768 

51 

1878 

1948 

2018 

2089 

2161 

2233 

2307 

2381 

2457 

2533 

2611 

2690 

2770 

52 

1879 

1949 

2019 

2090 

2162 

2235 

2308 

238o 

2458 

2535 

2612 

2691 

2771 

53 

1880 

1950 

2020 

2091 

2163 

2236 

2309 

2384 

2459 

2536 

2614 

2692 

2772 

54 

1881 

1951 

2021 

2092 

2164 

2237 

2311 

2385 

2461 

2537 

2615 

2694 

2774 

55 

1883 

1952 

2022 

2094 

2165 

2238 

2312 

2386 

2462 

2538 

2616 

2695 

2775 

5G 

1884 

1953 

2024 

2095 

2167 

2239 

2313 

238h 

2463 

2540 

2617 

2696 

2776 

57 

1885 

1955 

20-25 

209G 

2168 

2241 

2314 

2389 

2464 

2641 

2619 

2698 

2778 

58 

1886 

1956 

2026 

2097 

2169 

2242 

2316 

2390 

2466 

2542 

2620 

2699 

2779 

59 

1887 

1957 

2027 

2098 

2170 

2243 

2317 

2391 

2467 

2544 

2621 

2700 

2780 

TABLE  IV.               Meridianal  Parts.                      97 

/ 

42° 

43° 

440 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

52° 

53° 

54° 

0 

2782 

2863 

2946 

3030 

3116 

3203 

3292 

3382 

3474 

3569 

3665 

3764 

3865 

1 

2783 

2864 

2947 

3031 

3117 

3204 

3293 

3384 

3476 

3570 

3667 

3766 

3866 

2 

2734 

2866 

2949 

3033 

3118 

3206 

3295 

3385 

3478 

3572 

3668 

3767 

3868 

3 

2786 

2867 

2960 

3034 

2120 

3207 

3296 

3387 

3479 

3573 

367C 

3769 

3870 

4 

2787 

2869 

2951 

3036 

3121 

3209 

3298 

3388 

3481 

3575 

3672 

3770 

3871 

6 

2788 

2870 

2953 

3037 

3123 

3210 

3299 

3390 

3482 

3577 

3673 

3772 

3873 

6 

2790 

2871 

2954 

3038 

3123 

3212 

3301 

3391 

3484 

3578 

3675 

3774 

3375 

7 

2791 

2873 

2956 

3040 

3126 

3213 

3302 

3393 

3485 

3580 

3677 

3775 

3877 

8 

2792 

2874 

2957 

3041 

3127 

3214 

3303 

3394 

3487 

3582 

3678 

3777 

3878 

9 

2794 

2875 

2958 

3043 

3129 

3216 

3305 

3396 

3488 

3583 

3680 

3779 

3880 

10 

2795 

2877 

2960 

3044 

3130 

3217 

3306 

3397 

3490 

3585 

3681 

3780 

3882 

11 

2797 

2878 

2961 

3046 

3131 

3219 

3308 

3399 

3492 

3586 

3683 

3782 

3883 

12 

2798 

2880 

2963 

3047 

3133 

3220 

3309 

3400 

3493 

3588 

3686 

3784 

3885 

13 

2799 

2881 

2964 

3048 

3134 

3222 

3311 

3402 

3495 

3590 

3686 

3785 

3887 

14 

2801 

2882 

2965 

3050 

3136 

3224 

3312 

3403 

3496 

3591 

3688 

3787 

3889 

15 

2802 

2884 

2967 

3051 

3137 

3225 

33141  3405 

3498 

3593 

3690 

3799 

3890 

16 

2803 

2885 

2968 

3053 

3139 

3226 

3316 

3407 

3499 

3594 

3691 

3790 

3892 

17 

2805 

2886 

2970 

3054 

3140 

3228 

3317 

3408 

3501 

3596 

3693 

3792 

3894 

18 

2806 

2888 

2971 

3055 

3142 

3229 

3319 

3410 

3503 

3598 

3695 

3794 

3895 

19 

2807 

2889 

2972 

3057 

3143 

3231 

3320 

3411 

3504 

3599 

3696 

3795 

3897 

20 

2809 

2891 

2974 

3058 

3144 

3232 

3322 

3413 

3506 

3601 

3698 

3797 

3899 

21 

2810 

2892 

2975 

3060 

3146 

3234 

3323 

3414 

3507 

3602 

3699 

3799 

3901 

22 

2811 

2893 

2976 

3061 

3147 

3235 

3325 

3416 

3609 

3604 

3701 

3800 

3902 

23 

2813 

2895 

2978 

3063 

3149 

3237 

3326 

3417 

3510 

3606 

3703 

3802 

3904 

24 

2814 

2896 

2979 

3064 

3150 

3238 

3328 

3419 

3512 

3607 

3704 

3804 

3906 

25 

2815 

2897 

2981 

3065 

3152 

3240 

3329 

3420 

3514 

3609 

3706 

3806 

3907 

26 

2817 

2899 

2982 

3067 

3153 

3241 

3331 

3422 

3515 

3610 

3708 

3807 

3909 

27 

2818 

2900 

2983 

3068 

3155 

3242 

3332 

3423 

3517 

3612 

3709 

3809 

3911 

28 

2820 

2902 

2985 

3070 

3166 

3244 

3334 

3425 

3518 

3614 

3711 

3811 

3913 

29 

2821 

2903 

2986 

3071 

3157 

3246 

3335 

3427 

3620 

3615 

3713 

3812 

3914 

30 

2822 

2904 

2988 

3073 

3159 

3247 

3337 

3428 

3521 

3617 

3714 

3814 

3916 

31 

2824 

2906 

2989 

3074 

3160 

3248 

3338 

3430 

3523 

3618 

3716 

3816 

3918 

32 

2825 

2907 

2991 

3076 

3162 

3250 

3340 

3431 

3525 

3620 

3717 

3817 

3919 

33 

2826 

2908 

2992 

3077 

3163 

3251 

3341 

3433 

3526 

3622 

3719 

3819 

3921 

34 

2828 

2910 

2993 

3078 

3165 

3253 

3343 

3434 

3528 

3623 

3721 

3821 

3923 

35 

2829 

2911 

2995 

3080 

3166 

3264 

3344 

3436 

3529 

3625 

3722 

3822 

3926 

36 

2830 

2913 

2996 

3081 

3168 

3256 

3346 

3437 

3531 

3626 

3724 

3824 

3926 

37 

2832 

2914 

2998 

3083 

3169 

3257 

3347 

3439 

3532 

3628 

3726 

3826 

3928 

38 

2833 

2915 

2999 

3084 

3171 

3259 

3349 

3440 

3534 

3630 

3727 

3827 

3930 

39 

2834 

2917 

3000 

3085 

3172 

3260 

3350 

3442 

3536 

3631 

3729 

3829 

3932 

40 

2836 

2918 

3002 

3087 

3173 

3262 

3352 

3543 

3537 

3633 

3731 

3831 

3933 

41 

2837 

2919 

3003 

3088 

3175 

3263 

3353 

3445 

3539 

3634 

3732 

3832 

3936 

42 

2839 

2921 

3005 

3090 

3176 

3265 

3355 

3447 

3540 

3636 

3734 

3834 

3937 

43 

2840 

2922 

2006 

3091 

3178 

3266 

3356 

3448 

3542 

3638 

3736 

3836 

3938 

44 

2841 

2924 

3007 

3093 

3179 

3268 

3358 

3450 

3543 

3639 

3737 

3838 

3940 

45 

2843 

2925 

3009 

3094 

3181 

3269 

3369 

3561 

3645 

3641 

3739 

3839 

3942 

46 

2844 

2926 

3010 

3095 

3182 

3271 

3361 

3453 

3547 

3643 

3741 

3841 

3944 

47 

2845 

2928 

3012 

3097 

3184 

3272 

3362 

3454 

3548 

3644 

3742 

3843 

3945 

48 

2847 

2929 

3013 

3098 

3185 

3274 

3364 

3456 

3550 

3646 

3744 

3844 

3947 

49 

2848 

2931 

3014 

3100 

3187 

3275 

3365 

3457 

3551 

3647 

3746 

3846 

3949 

50 

2849 

2932 

3016 

3101 

3188 

3277 

3367 

3459 

3553 

3649 

3747 

3848 

3951 

51 

2851 

2933 

3017 

3103 

3190 

3278 

3368 

3460 

3555 

3651 

3749 

3849 

3952 

52 

2852 

2935 

3019 

3104 

3191 

3280 

3370 

3462 

3556 

3652 

3760 

3851 

3954 

53 

2854 

2936 

3020 

3105 

3192 

3281 

3371 

3464 

3558 

3654 

3752 

3853 

3956 

54 

2855 

2937 

3021 

3107 

3194 

3283 

3373 

3465 

3559 

3665 

3754 

3854 

3958 

55 

2856 

2939 

3023 

3108 

3196 

3284 

3374 

3467 

3561 

3667 

3766 

3856 

3959 

56 

2868 

2940 

3024 

3110 

3197 

3286 

3376 

3468 

3562 

3659 

3767 

3858 

3961 

57 

2859 

2942 

3026 

3111 

3198 

3287 

3378 

3470 

3564 

3660 

3769 

3860 

3963 

58 

2860 

2943 

3027 

3113 

3200 

3289 

3379 

3471 

3566 

3662 

3760 

3861 

3964 

69  2862 

2944 

3029 

3114 

3201 

3290 

3381 

3473 

3667 

3664 

3762 

3863 

3966 

98                     Meridianal  Parte.              TABLE  IV. 

/ 

55° 

56° 

57° 

68° 

59° 

60° 

61° 

62° 

63° 

64° 

65° 

66° 

67° 

~o 

3968 

4074 

4183 

4294 

4409 

4527 

4649 

4776 

4906 

5039 

6179 

6324 

6474 

i 

3970 

4076 

4184 

4296 

4411 

4529 

4661 

4777 

4907 

6042 

6181 

5326 

5477 

2 

3971 

4077 

4186 

4298 

4413 

4531 

4653 

4779 

4909 

6044 

6184 

5328 

6479 

3 

3973 

4079 

4188 

4300 

4415 

4533 

4655 

4781 

4912 

6046 

6186 

6331 

5482 

4 

3975 

4081 

4190 

4302 

4417 

4535 

4657 

4784 

4914 

5049 

6188 

6333 

6484 

6 

3977 

4083 

4192 

4304 

4419 

4537 

4660 

4786 

4916 

6051 

6191 

6336 

6487 

6 

3978 

4085 

4194 

4306 

4421 

4539 

4662 

4788 

4918 

5053 

6193 

6338 

5489 

7 

3980 

4086 

4195 

4308 

4423 

4541 

4664 

4790 

4920 

6055 

5195 

6341 

6492 

8 

3982 

4058 

4197 

4309 

4425 

4543 

4666 

4792 

4923 

5058 

6198 

5343 

5495 

9 

3984 

4080 

4199 

4311 

4427 

4545 

4668 

4794 

4925 

6060 

5200 

6346 

5497 

10 

3985 

4092 

4202 

4313 

4429 

4547 

4670 

4796 

4927 

5062 

5203 

6348 

5500 

11 

3987 

4094 

4203 

4315 

4431 

4549 

4672 

4798 

4929 

6065 

6205 

5351 

5502 

12 

3989 

4095 

4205 

4317 

4433 

4551 

4674 

4801 

4931 

6067 

6207 

5363 

6505 

13 

3991 

4097 

4207 

4319 

4434 

4553 

4676 

4805 

4934 

5069 

5210 

6356 

5607 

14 

3992 

4099 

4208 

4321 

4436 

4555 

4678 

4808 

4936 

6071 

5212 

5358 

5510 

15 

3994 

4101 

4210 

4323 

4438 

4557 

4680 

4807 

4938 

6074 

6214 

5361 

6513 

16 

3996 

4103 

4212 

4325 

4440 

4559 

4682 

4809 

4940 

5076 

6217 

5363 

6515 

17 

3998 

4104 

4214 

4327 

4442 

4562 

4684 

4811 

4943 

6078 

5219 

6366 

5518 

18 

3999 

4106 

4216 

4328 

1111 

4564 

4687 

4814 

4945 

5081 

6222 

KOCQ 

6520 

19 

4001 

4108 

4218 

4330 

4446 

4566 

4689 

4816 

4947 

6083 

6224 

oooo 
5371 

5523 

20 

4003 

4110 

4220 

4332 

4448 

4568 

4691 

4818 

4949 

6085 

5226 

6373 

6526 

21 

4005 

4112 

4221 

4334 

4450 

4570 

4693 

4820 

4951 

6088 

5229 

6376 

5528 

22 

4006 

4113 

4223 

4336 

4452 

4572 

4695 

4822 

4954 

5090 

5231 

5378 

5531 

23 

4008 

4115 

4225 

4338 

4454 

4574 

4697 

4824 

4956 

6092 

6234 

5380 

5533 

24 

4010 

4117 

4227 

4340 

4456 

4576 

4699 

4826 

4958 

5095 

6236 

6383 

6536 

25 

4012 

4119 

4229 

4342 

4458 

4578 

4701 

4829 

4960 

5097 

6238 

6385 

5539 

26 

4014 

4121 

4231 

4344 

4460 

4580 

4703 

4831 

4963 

6099 

5241 

5388 

6541 

27 

4015 

4122 

4232 

4346 

4462 

4582 

4705 

4833 

4965 

5102 

6243 

5390 

6544 

28 

4017 

4124 

4234 

4347 

4464 

4584 

4707 

4835 

4967 

5104 

5246 

5393 

5546 

29 

4019 

4126 

4236 

4349 

4466 

4586 

4710 

4837 

4969 

5106 

5248 

5395 

5549 

30 

4021 

4128 

4238 

4351 

4468 

4588 

4712 

4839 

4972 

5108 

6250 

5398 

5552 

31 

4022 

4130 

4240 

4353 

4470 

4590 

4714 

4842 

4974 

6111 

6253 

5401 

5564 

32 

4024 

4132 

4242 

4355 

4472 

4592 

4716 

4844 

4976 

6113 

6255 

6403 

5557 

33 

4026 

4133 

4244 

4357 

4474 

4594 

4718 

4846 

4978 

6115 

6258 

6406 

5559 

34 

4028 

4135 

4246 

4359 

4476 

4596 

4720 

4848 

4981 

5118 

5260 

6408 

5562 

35 

4029 

4137 

4247 

4361 

4478 

4598 

4722 

4850 

4983 

5120 

6263 

6411 

5565 

36 

4031 

4139 

4249 

4363 

4480 

4600 

4724 

4862 

4986 

6122 

5265 

5413 

5567 

37 

4033 

4141 

4251 

4366 

4482 

4602 

4726 

4865 

4987 

6125 

6267 

5416 

6570 

38 

4035 

4142 

4253 

4367 

4484 

4604 

4728 

4857 

4990 

6127 

5270 

6418 

6573 

39 

4037 

4144 

4255 

4369 

4486 

4606 

4731 

4859 

4992 

5129 

6272 

5421 

6576 

40 

4038 

4146 

4257 

4370 

4488 

4608 

4733 

4861 

4994 

5132 

6275 

6423 

5578 

41 

4040 

4148 

4259 

4372 

4490 

4610 

4735 

4863 

4996 

5134 

5277 

5426 

6580 

42 

4042 

4150 

4250 

4374 

4492 

4612 

4736 

4865 

4999 

5136 

6280 

6428 

5583 

43 

4044 

4152 

4262 

4376 

4494 

4614 

4739 

4868 

5001 

6139 

6282 

5431 

6586 

44 

4045 

4153 

4264 

4378 

4495 

4616 

4741 

4870 

5003 

5141 

5284 

6433 

5588 

45 

4047 

4155 

4266 

4380 

4497 

4618 

4743 

4872 

5005 

5143 

6287 

6436 

5591 

46 

4049 

4157 

4268 

4382 

4499 

4620 

4745 

4874 

5008 

6146 

5289 

6438 

5594 

47 

4051 

4159 

4270 

4384 

4501 

4623 

4747 

4876 

5010 

5148 

5292 

5441 

5598 

48 

4052 

4161 

4272 

4386 

4503 

4625 

4750 

4879 

5012 

6151 

6294 

5443 

5599 

49 

4054 

4162 

4274 

4388 

4505 

4627 

4762 

4881 

5014 

6163 

6297 

6446 

5602 

50 

4056 

4164 

4275 

4390 

4507 

4629 

4754 

4883 

5017 

5155 

5299 

6448 

5604 

51 

4058 

4166 

4277 

4392 

4509 

4631 

4766 

4885 

5019 

5158 

5301 

6451 

5607 

52 

4060 

4168 

4279 

4394 

4511 

4633 

4758 

4887 

5021 

5160 

5304 

6454 

6610 

53 

4061 

4170 

4281 

4396 

4513 

4636 

4760 

4890 

5023 

6162 

5306 

6466 

6612 

54 

4063 

4172 

4283 

4398 

4515 

4637 

4762 

4892 

6026 

5165 

6309 

5458 

6615 

55 

4065 

4173 

4285 

4399 

4517 

4639 

4764 

4894 

5028 

5167 

6311 

6461 

6617 

56 

4067 

4175 

4287 

4401 

4519 

4641 

4766 

4896 

6030 

5169 

5314 

6464 

6620 

57 

4069 

4177 

4289 

4403 

4521 

4643 

4769 

4898 

6033 

6172 

5316 

5466 

6623 

58 

4070 

4179 

4291 

4405 

4523 

4646 

4771 

4901 

6035 

5174 

5319 

5469 

6625 

59 

4072 

4181 

4292 

4407 

4525 

4647 

4773 

4903 

6037 

5176 

6321 

6471 

5628 

TABLE  IV.               Meridianal  Parts.                      99 

' 

68° 

69° 

70° 

71° 

72° 

73° 

74° 

75° 

76° 

77°  |  78° 

79° 

80° 

0 

5631 

5795 

6966 

6146 

6335 

6534 

6746 

6970 

7210 

7467 

7745 

8046 

8375 

1 

5633 

5797 

6969 

6149 

6338 

6638 

6749 

6974 

7214 

7472 

7749 

8051 

8381 

2 

5636 

5800 

5972 

6152 

6341 

6541 

6753 

6978 

7218 

7476 

7754 

8056 

8387 

3 

5639 

5803 

5975 

6155 

6345 

6545 

6757 

6982 

7222 

7481 

7759 

8061 

8393 

4 

5642 

5806 

5978 

6158 

6348 

6548 

6760 

6986 

7227 

7486 

7764 

8067 

8398 

6 

5644 

5809 

5981 

6161 

6351 

6552 

6764 

6990 

7231 

7490 

7769 

8072 

8404 

6 

5646 

5811 

5984 

6164 

6354 

6555 

6768 

6994 

7235 

7494 

7774 

8077 

8410 

7 

5650 

6814 

5986 

6167 

6358 

6658 

6771 

6997 

7239 

7498 

7778 

8083 

8416 

8 

5652 

5817 

5989 

6170 

6361 

6562 

6775 

7001 

7243 

7603 

7783 

8088 

8422 

9 

5655 

5820 

5992 

6173 

6364 

6565 

6779 

7005 

7247 

7507 

7788 

8093 

8427 

10 

5658 

5823 

5995 

6177 

6367 

6569 

6782 

7009 

7252 

7612 

7793 

8099 

8433 

11 

5660 

5826 

5998 

6180 

6371 

6572 

6786 

7013 

7256 

7516 

7798 

8104 

8439 

12 

5663 

6828 

6001 

6183 

6374 

6576 

6790 

7017 

7260 

7521 

7803 

8109 

8445 

13 

5666 

5831 

6004 

6186 

6377 

6579 

6793 

7021 

7264 

7525 

7808 

8115 

8451 

14 

5668 

6834 

6007 

6189 

6380 

9583 

6797 

7025 

7268 

7630 

7813 

8120 

8457 

15 

5671 

5837 

6010 

6192 

6384 

6586 

6801 

7029 

7273 

7535 

7817 

8125 

8463 

16 

5674 

6839 

6013 

6195 

6387 

6590 

6804 

7033 

7277 

7439 

7821 

8131 

8469 

17 

5676 

5842 

6016 

6198 

6390 

6593 

6808 

7027 

7281 

7544 

7827 

8136 

8474 

18 

5679 

5845 

6019 

6201 

6394 

6597 

6812 

7041 

7286 

7548 

7832 

8141 

8480 

19 

5682 

5848 

6022 

6205 

6397 

6600 

6815 

7045 

7289 

7553 

7837 

8147 

8486 

20 

5685 

5851 

6025 

6208 

6400 

6603 

6819 

7048 

7294 

7557 

7842 

8152 

8492 

21 

5687 

5854 

6028 

6211 

6403 

5607 

6823 

7052 

7298 

7562 

7847 

8158 

8498 

22 

5690 

5856 

6031 

6214 

6407 

6610 

6826 

7056 

7302 

7566 

7852 

8163 

8504 

23 

5693 

5859 

6034 

6217 

6410 

6614 

6830 

7060 

7306 

7571 

7857 

8168 

8610 

24 

5695 

6862 

6037 

6220 

6413 

6617 

6834 

7064 

7311 

7676 

7862 

8174 

8516 

25 

5698 

5865 

6040 

6223 

6417 

6621 

6838 

7068 

7315 

7580 

7867 

9179 

8522 

26 

5701 

6868 

6043 

6226 

6420 

6624 

6841 

7072 

7319 

7685 

7872 

8186 

8628 

27 

6704 

5871 

6046 

6230 

6423 

6628 

6845 

7076 

7323 

7589 

7877 

8190 

8634 

28 

5706 

5874 

6049 

6233 

6427 

6631 

6849 

7080 

7328 

7594 

7882 

8196 

8540 

29 

5709 

5876 

6052 

6236 

6430 

6635 

6853 

7084 

7332 

7699 

7887 

8201 

8546 

30 

5712 

6879 

6055 

6239 

6433 

6639 

6856 

7088 

7336 

7603 

7892 

8207 

8552 

31 

5715 

6882 

6058 

6242 

6437 

6642 

6860 

7092 

7341 

7608 

7897 

8212 

8558 

32 

5717 

6885 

6061 

6245 

6440 

6646 

6864 

7096 

7345 

7612 

7902 

8218 

8565 

33 

5720 

6888 

6064 

6249 

6443 

6649 

6868 

7100 

7349 

7617 

7907 

8223 

8571 

34 

6723 

5891 

6067 

6252 

6447 

6653 

6871 

7104 

7353 

7622 

7912 

8229 

8577 

35 

5725 

5894 

6070 

6265 

6450 

6656 

6876 

7108 

7358 

7626 

7917 

8234 

8683 

36 

5728 

6896 

6073 

6268 

6453 

6660 

6879 

7112 

7362 

7631 

7922 

8240 

8589 

37 

6731 

5899 

6076 

6261 

6457 

6663 

6883 

7116 

7366 

7636 

7927 

8246 

8595 

38 

6734 

6902 

6079 

6264 

6460 

6667 

6886 

7128 

7371 

7640 

7932 

8251 

8601 

39 

5736 

5905 

6082 

6268 

6463 

6670 

6890 

7124 

7375 

7645 

7937 

8256 

8607 

40 

5739 

5908 

6085 

6271 

6467 

6674 

6894 

7128 

7379 

7650 

7942 

8262 

8614 

41 

5742 

5911 

6088 

6274 

6470 

6677 

6898 

7132 

7384 

7654 

7948 

8267 

8620 

42 

6745 

6914 

6091 

6277 

6473 

6681 

6901 

7136 

7388 

7659 

7953 

8273 

8626 

43 

5747 

5917 

6094 

6280 

6477 

6685 

6905 

7140 

7392 

7664 

7958 

8279 

8632 

44 

5750 

5919 

6097 

6283 

6480 

6688 

6909 

7145 

7397 

7668 

7963 

8284 

8638 

45 

5753 

5922 

6100 

6287 

6483 

6692 

6913 

7149 

7401 

7673 

7968 

8290 

8644 

46 

5756 

5925 

6103 

6290 

6487 

6695 

6917 

7153 

7406 

7678 

7973 

8295 

8651 

47 

5758 

5928 

6106 

9293 

6490 

6699 

6920 

7167 

7410 

7683 

7978 

8301 

8667 

48 

5761 

5931 

6109 

6296 

6494 

6702 

6924 

7161 

7414 

7687 

7983 

8307 

8663 

49 

5764 

5934 

6112 

6299 

6497 

6706 

6928 

7165 

7419 

7692 

7989 

8312 

8669 

50 

5767 

5937 

6115 

6303 

6500 

6710 

6932 

7169 

7423 

7697 

7994 

8318 

8676 

51 

5770 

6940 

6118 

6306 

6504 

6713 

6936 

7173 

7427 

7702 

7999 

8324 

8682 

52 

5772 

5943 

6121 

6309 

6507 

6717 

6940 

7177 

7432 

7706 

8004 

8329 

8688 

63 

6775 

6946 

6124 

6312 

6511 

6720 

6943 

7181 

7436 

7711 

8009 

8335 

8696 

54 

5778 

5948 

6127 

6315 

6514 

6724 

6947 

7185 

7441 

7716 

8914 

8341 

8701 

55 

5781 

6951 

6130 

6319 

6517 

6728 

6951 

7189 

7446 

7721 

8020 

8347 

8707 

56 

5783 

6954 

6133 

6322 

6521 

6731 

6965 

7194 

7449 

7725 

8025 

8352 

8714 

57 

6786 

5957 

6136 

6325 

6624 

6735 

6969 

7198 

7464 

7730 

8030 

8358 

8720 

58 

5789 

6960 

6140 

6328 

6528 

6738 

6963 

7202 

7458 

7735 

8035 

8364 

8726 

59 

5792 

6963 

6143 

6332 

6631 

6742 

6966 

7206 

7463 

7740 

8040 

8369 

8733 

100                     Meridianal  Parts.               TABLE  IV. 

' 

81° 

82° 

83° 

84° 

85° 

"o 

8739 

9145 

"9606 

10137 

10766 

1 

8745 

9153 

9614 

10146 

10776 

o 

8762 

9160 

9622 

10156 

10788 

3 

8758 

9167 

9631 

10166 

10799 

4 

8765 

9174 

9639 

10175 

10811 

5 

8771 

9182 

9647 

10185 

10822 

6 

8778 

9189 

9655 

10195 

10834 

7 

8784 

9197 

9664 

10205 

10846 

8 

8791 

9203 

9672 

10214 

10858 

„ 

9 

8797 

9211 

9683 

10224 

10869 

10 

8804 

9218 

9689 

10234 

10881 

11 

8810 

9225 

9697 

10244 

10893 

12 

8817 

9233 

9706 

10264 

10905 

13 

8823 

9240 

9714 

10364 

10917 

14 

8830 

9248 

9723 

10273 

10929 

15 

8836 

9255 

9731 

10283 

10941 

16 

8843 

9262 

9740 

10293 

10953 

- 

17 

8849 

9270 

9748 

10303 

10965 

18 

8856 

9277 

9757 

10314 

10978 

19 

8863 

9285 

9765 

10324 

10990 

20 

8869 

9292 

9774 

10334 

11002 

21 

8876 

9300 

9783 

10344 

11014 

22 

8883 

9307 

9791 

10354 

11027 

23 

8889 

9315 

9800 

10364 

11039 

24 

8896 

9322 

9809 

10374 

11052 

25 

8903 

9330 

9817 

10386 

11064 

26 

8909 

9337 

9826 

10396 

11077 

27 

8916 

9345 

9835 

10405 

11089 

28 

8923 

9353 

9844 

10416 

11102 

29 

8930 

9360 

9852 

10426 

11115 

30 

8936 

9368 

9861 

10437 

11127 

31 

8943 

9376 

9870 

10447 

11140 

32 

8950 

9383 

9879 

10457 

11153 

33 

8957 

9391 

9888 

10468 

11166 

34 

8963 

9399 

9897 

10479 

11179 

35 

8970 

9407 

9906 

10489 

11192 

36 

8977 

9414 

9915 

10500 

11205 

37 

8984 

9422 

9924 

10510 

11218 

38 

8991 

9430 

9933 

10521 

11231 

39 

8998 

9438 

9942 

10532 

11244 

40 

9005 

9445 

9951 

10542 

11257 

41 

9012 

9453 

9960 

10563 

11270 

42 

9018 

9461 

9969 

10664 

11284 

43 

9025 

9469 

9978 

10575 

11297 

44 

9032 

9477 

9987 

10586 

11310 

45 

9039 

9485 

9996 

10597 

11324 

46 

9046 

9493 

10005 

10608 

11337 

47 

9053 

9501 

10015 

10619 

11361 

48 

9060 

9509 

10024 

10630 

11365 

49 

9067 

9517 

10033 

10641 

11378 

50 

9074 

9525 

10043 

10652 

11392 

51 

9081 

9533 

10052 

10663 

11406 

52 

9088 

9541 

10061 

10674 

11420 

53 

9096 

9549 

10071 

10686 

11434 

54 

9103 

9557 

10080 

10696 

11448 

55 

9110 

9565 

10089 

10708 

11462 

50 

9117 

9573 

10099 

10719 

11476 

57 

9124 

9581 

10108 

10730 

11490 

' 

58 

9131 

9589 

10118 

10742 

11504 

59 

9138 

9598 

10127 

10763 

11518 

TABLE 

V. 

TABLE  VII.              mi 

Dip  of 

the  Sea  Horizon.          Mean  Refraction  of  Celestial  Objects. 

wa 

hH 

9 

V) 

u 

w 

5 

Al 

Rcfr. 

Alt 

Kef 

Al 

Ref 

Alt 

Refr 

Alt 

Kefr 

er! 

2.o 

§_ 

SI 

|o 

1 

0 

0 

33  C 

o 

10  0 

5  1 

o 
20 

2 

0 

32 

1  30 

o 
67 

n 
24 

22, 

t 

j1 

3° 

* 

1 

31  32 

10 

5  1 

1 

2 

4 

1  29 

68 

23 

— 

— 

__ 

2 

29  50 

20 

5  0 

2 

2 

33 

1  28 

69 

22 

o 

59 

38 

u 

3 

28  23 

30 

5  0 

3 

2 

2 

1  26 

70 

21 

2 

I 

^ 

41 

^, 

4 

27  00 

40 

4  6 

4 

2 

4 

1  25 

71 

19 

3 
4 
5 
6 

7 

1 

1 

2 
2 

2 

4'2 
oh 
12 
25 
36 

TrJL 

44 
47 
60 
63 
66 

G 
G 
G 

7 

7 

32 
45 
58 
10 
12 

6 
1 
1 

2 
3 

25  42 
24  29 
23  20 
22  15 
21  15 

50 
11  0 
10 
20 
30 

4  5 
4  4 
4  4 
4  3 
4  3 

6 
21 
1 

2 
3 

2 
2 
2 
2 
2 

34 
2 
4 
35 

2 

1  24 
1  23 
1  22 
1  21 
1  20 

72 
73 
74 
75 
76 

18 
17 
16 
15 
14 

8 

2  47 

69 

7 

24 

4 

20  1 

40 

4  3 

4 

2  2 

4 

1  19 

77 

13 

9 

2  57 

62 

7 

45 

5 

19  2 

50 

4  2 

6 

2  2 

36 

1  18 

78 

12 

10 

3  07 

66 

7 

66 

2 

18  3 

12  0 

4  2 

22 

2  2 

3 

17 

79 

11 

11 

3 

1G 

68 

8 

07 

1 

17  4 

10 

4  2 

1 

2 

37 

16 

80 

10 

12 

3  25 

71 

8 

18 

2 

17  0 

20 

4  1 

2 

2 

3 

14 

81 

9 

13 

3  33 

74 

8 

28 

3 

16  2 

30 

4  1 

3 

2 

38 

13 

82 

8 

14 

3  41 

77 

8 

38 

4 

15  4 

40 

4  0 

4 

2 

3 

11 

83 

7 

15 

3  49 

80 

8 

48 

5 

15  0 

50 

4  0 

5 

2 

39 

10 

34 

6 

16 

3  56 

83 

8 

68 

3 

14  3 

13  0 

4  0 

23 

2 

3 

1  09 

85 

6 

17 

4  04 

86 

9 

08 

1 

14  0 

10 

4  0 

10 

2  1 

40  0 

1  08 

86 

4 

18 

4 

11 

i 

i9 

9 

17 

2 

13  34 

20 

3  6 

20 

2  12 

3 

1  07 

87 

3 

19 

4  17 

92 

9 

26 

3 

13  06 

30 

3  64 

30 

2  1 

41  0 

1  05 

88 

2 

20 

4  24 

95 

9 

36 

40 

12  40 

49 

3  6 

40 

2  10 

30 

1  04 

89 

1 

21 

4  31 

< 

)8 

9 

46 

50 

12  15 

60 

3  48 

60 

2  09 

42  0 

1  03 

90 

0 

22 

4  37 

101 

9 

64 

4  0 

11  61 

14  0 

3  45 

24  0 

2  08 

30 

1  02 

23 

4  43 

104 

10 

02 

10 

11  29 

10 

3  43 

10 

2  07 

43  0 

1  01 

24 

4  49 

107 

10 

11 

20 

11  08 

20 

3  40 

20 

2  06 

30 

1  00 

26 

4  55 

110  10 

19 

30 

10  48 

30 

3  38 

30 

2  05 

44  0 

0  59 

26 

27 

5  01 

5  07 

113  10 

lie!  10 

28 
36 

40 
50 

10  29 
10  11 

40 
60 

3  35 
3  33 

40 
50 

2  04 
2  03 

80 
45  0 

0  58 
0  67 

28 

5 

.T 

11 

9  in 

44 

29 

6  18 

122 

10 

62 

S  ( 

9  54 

15  0 

3  30 

25  0 

2  02 

30 

0  66 

30 

6  24 

125 

1  1 

00 

10 

9  38 

10 

3  28 

10 

2  01 

46  0 

0  55 

31 
32 
33 
34 
35 

6  29 
6  34 
6  39 
5  44 
6  49 

128  11 
131  11 
134  11 
137  11 
140!  11 

08 
16 
24 
31 
39 

20 
30 
40 
50 
6  0 
10 

9  23 
9  08 
8  64 
8  41 
8  28 
8  15 

20 
30 
40 
60 
16  ( 

3  26 
3  24 
3  21 
3  19 
3  17 
3  16 

20 
30 
40 
60 
26  0 

2  00 
i  59 
1  68 
1  67 
66 

30 
47  0 
30 
48  0 
30 
Q  0 

0  64 
0  63 
0  62 
61 
60 
49 

20 

8  03 

2( 

3  12 

2( 

.  55 

:«7  I 

30 

49 

TABLE  VI. 

30 

7  16 

30 

3  10 

30 

54 

50  0 

48 

Dip  of  the  Sea  Horizon  at 
different  Distances  from  it. 

40 
50 
7  0 

7  40 
7  30 
7  20 
71  1 

40 
50 
17  C 

3  08 
3  06 
3  04 

q  AC 

40 
60 
27  0 
15 

53 
52 
51 
fif 

30 
51  0 
30 

2r 

47 
46 
45 

Dist. 

Hight  of  Eye  in  Ft. 

2( 

J.  1 

7  02 

2C 

ij  Uc 

3  01 

3C 

ot 
49 

\ 

3C 

44 

in 

Miles. 

5 

10 

15 

2( 

)  25 

30 

30 

Af\ 

6  5J 

3C 

2  69 

45 

O  A 

'.  48 

3  C 

43 

j 

11 

2'2 

34 

4f 

5G 

i 
68 

41 

50 

6  46 
6  37 

4C 
50 

2  57 
2  65 

o  0 
16 

47 
46 

30 
4  0 

42 
41 

i 

G 

4 

11 

8 

17 

12 

22 
15 

28 
19 

34 
23 

8  0 
10 

6  29 
6  22 

8  0 
10 

2  64 
2  62 

30 
45 

45 
44 

5  0 
6  0 

40 
38 

1* 

4 

G 

9 

12 

15 

17 

20 

6  15 

20 

2  61 

9  0 

42 

7  0 

37 

U 

3 

5 

7 

9 

12 

14 

30 

6  08 

30 

2  49 

20 

41 

8  0 

35 

1* 

3 

4 

6 

8 

9 

12 

40 

6  01 

40 

2  47 

40 

40 

9  0 

34 

2 

2 

3 

5 

G 

8 

10 

60 

6  65 

60 

2  46 

0  0 

38 

0  0 

33 

Si 

2 

3 

5 

G 

7 

8 

9  0 

5  98 

9  0 

2  44 

20 

37 

1  0 

32 

3 

2 

3 

4 

5 

6 

7 

10 

6  42 

10 

2  43 

40 

36 

2  0 

30 

8* 

2 

3 

4 

5 

G 

6 

20 

5  46 

20 

I  41 

1  0 

36 

3  0 

29 

4 

2 

3 

4 

4 

5 

6 

30 

5  41 

30 

I  40 

20 

33 

4  0 

28 

6 

2 

3 

4 

4 

5 

5 

40 

6  25 

40 

38 

40 

32 

5  0 

26 

6 

2 

3 

4 

4 

5 

6 

50 

6  20 

60  37 

2  0 

31 

6  0 

25 

-,-  -;*—  ^ 

I 


M1893 

£-45 


TC   (3543 


f 


